The Pennsylvania State University
The Graduate School
Eberly College of Science
AN ASYMPTOTIC MUKAI MODEL OF M6
A Dissertation in
Mathematics
by
Evgeny Mayanskiy
c 2013 Evgeny Mayanskiy
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
August 2013
The dissertation of Evgeny Mayanskiy was reviewed and approved∗ by the following:
Yuri Zarhin
Professor of Mathematics
Dissertation Advisor, Chair of Committee
Robert Vaughan
Professor of Mathematics
Dale Brownawell
Distinguished Professor of Mathematics
Karl Schwede
Assistant Professor of Mathematics
Murat Gunaydin
Professor of Physics
Svetlana Katok
Professor of Mathematics
Chair of Graduate Program
∗
Signatures are on file in the Graduate School.
Abstract
We study the Mukai construction of a general curve of genus 6 as a complete intersection of the Grassmannian of lines in P4 with a codimension 5 quadric in the Plücker
space. We formulate the relevant GIT problem in general and then solve it for the
large values of the GIT parameter. This allows us to conclude that asymptotically
Mukai compact model of M6 parametrizes double anticanonical curves on the smooth
del Pezzo surface of degree 5. As a byproduct of our study we obtain an explicit
geometric interpretation of Ozeki classification of orbits of a certain prehomogeneous
space. This complements earlier results of J.A. Todd [17].
iii
Table of Contents
List of Figures
vi
Acknowledgments
vii
Chapter 1
Introduction
1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2
Chapter 2
The parameter space
3
Chapter 3
Hilbert-Mumford criterion
10
Chapter 4
The asymptotic case
21
Appendix A
Classification of codimension 4 linear sections of the Grassmannian of lines in P4
32
Appendix B
Grassmannian degenerations of the del Pezzo quintic threefold
42
Appendix C
Excel macros which computes extremal destabilizing 1-PS
47
iv
Appendix D
Excel macros which identifies minimal sets of vanishing Plücker
coordinates for unstable points
55
Appendix E
MATLAB code which checks if a given matrix lies in the general
orbit of GL(4) × SL(5)
60
Bibliography
63
v
List of Figures
2.1
Two families. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1
A.2
A.3
A.4
A.5
A.6
A.7
Codimension
Codimension
Codimension
Codimension
Codimension
Codimension
Codimension
4
4
4
4
4
4
4
sections
sections
sections
sections
sections
sections
sections
of
of
of
of
of
of
of
G(2, 5) ⊂ P9 ,
G(2, 5) ⊂ P9 ,
G(2, 5) ⊂ P9 ,
G(2, 5) ⊂ P9 ,
G(2, 5) ⊂ P9 ,
G(2, 5) ⊂ P9 ,
G(2, 5) ⊂ P9 ,
orbits
orbits
orbits
orbits
orbits
orbits
orbits
1-5. . .
6-11. .
12-17.
18-22.
23-28.
29-35.
36-38.
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35
36
37
38
39
40
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B.1
B.2
B.3
B.4
Codimension
Codimension
Codimension
Codimension
3
3
3
3
sections
sections
sections
sections
of
of
of
of
G(2, 5) ⊂ P9 ,
G(2, 5) ⊂ P9 ,
G(2, 5) ⊂ P9 ,
G(2, 5) ⊂ P9 ,
orbits
orbits
orbits
orbits
1-5. . .
6-10. .
11-14.
15-21.
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4
Acknowledgments
Our work was made possible because of the Teaching Assistantship support from the
Mathematical Department of the Pennsylvania State University.
This work is a part of a larger project (joint with Damiano Fulghesu), where we
study Mukai models of moduli spaces of curves of low genera and their relations with
the Hassett-Keel program. We started working on it during the AIM Workshop ’Log
minimal model program for moduli spaces’ in December 2012. We thank organizers
of the workshop and staff of the American Institute of Mathematics for providing
excellent working conditions and especially Brendan Hassett, who pointed out this
problem during the workshop.
Two months after we obtained the results presented in this thesis an entirely
independent work by Fabian Müller appeared1 which identified our asymptotic GIT
quotient with the final log canonical model of M6 .
1
Fabian Müller, The final log canonical model of M6 , arxiv: 1303.6843 (2013).
vii
Chapter
1
Introduction
1.1
Preface
This thesis contains the beginnings of the study of Mukai models of the moduli space
of curves of genus 6. Mukai-like constructions of curves, surfaces, threefolds... as complete intersections and sections of Grassmannians suggest that Geometric Invariant
Theory can be used in order to construct quasiprojective moduli spaces of such varieties as well as their natural compactifications. Nevertheless, it appears that this
application of Mumford’s GIT did not attract much attention and was not explored
much. We would like to point out [2], [3], [4], [7] among other works in this direction.
In this thesis we study one such GIT construction of a moduli space of curves of
genus 6. We hope that the methods we use and observations we make will be useful
for other analogous GIT constructions involving sections of Grassmannians.
As a byproduct of our study we obtain an explicit geometric classification of
codimension 4 linear sections of the Grassmannian of lines in P4 . Linear sections of
Grassmannians have a very beautiful geometry (see [11], [16], [15], [8], [6], [1] among
many others). Our result is an elementary consequence of the Ozeki classification
of orbits of a certain prehomogeneous space [14]. The problem of classifying linear
sections (in particular, linear sections of codimensions 3 and 4) of the Grassmannian
of lines in P4 was addressed earlier by J.A. Todd [17]. We believe that our results
complement his study.
2
1.2
Statement of the problem
Let us recall the following result of Mukai (see section 5 of [12]).
Theorem. (Mukai, [12]) Let C be a general curve of genus 6. Then:
(1) there exists a unique stable rank 2 vector bundle Fmax on C such that
det(Fmax ) = ωC
is the canonical bundle and h0 (C, Fmax ) = 5 is maximal possible,
(2) Fmax is generated by global sections,
(3) the morphism C → G(2, H 0 (C, Fmax )) defined by the global sections of Fmax is
a closed embedding which represents C as a complete intersection of
G(2, H 0 (C, Fmax ))
V
with a codimension 5 quadric in P( 2 H 0 (C, Fmax )) (where the Grassmannian
is embedded via the Plücker embedding).
Vice versa, it is easy to see that a general such complete intersection is a canonical curve of genus 6.
This suggests the following natural question.
Problem. Describe the quotient of a compactification of a parameter space of
V
complete intersections G(2, V ) ∩ (codim 5 quadric in P( 2 V )) by the automorphism
group Aut(V ) and represent this quotient as a compact model of M6 .
Chapter
2
The parameter space
In this chapter we define a compactification X of a parameter space of complete
intersections as above.
Let k = C and V = k 5 with the standard action of G = SL(5). Let G(2, V ) ,→
V
P( 2 V ) be the Plücker embedding of the Grassmannian of affine 2-planes in V .
We recall that the ideal IP of G(2, V ) as a subvariety of P(
V2
V ) can be generated
by quadratic polynomials
p1 = Z0 Z7 − Z1 Z5 + Z2 Z4 , p2 = Z0 Z8 − Z1 Z6 + Z3 Z4 , p3 = Z0 Z9 − Z2 Z6 + Z3 Z5 ,
p4 = Z1 Z9 − Z2 Z8 + Z3 Z7 ,
p5 = Z4 Z9 − Z5 Z8 + Z6 Z7 ,
where Z0 , Z1 , Z2 , Z3 , Z4 , Z5 , Z6 , Z7 , Z8 , Z9 are the homogeneous coordinates on
^2
P(
V)∼
= P9 .
If X1 , X2 , X3 , X4 , X5 are coordinates of V , then one can take
Z 0 = X1 ∧ X2 ,
Z1 = X1 ∧ X3 ,
Z2 = X1 ∧ X4 ,
Z3 = X1 ∧ X5 ,
Z4 = X2 ∧ X3 ,
Z5 = X2 ∧ X4 ,
Z6 = X2 ∧ X5 ,
Z7 = X3 ∧ X4 ,
Z8 = X3 ∧ X5 ,
Z9 = X4 ∧ X5 .
Consider the Grassmannian G(4, H 0 (G(2, V ), OG(2,V ) (1))) = G(4,
V2
V ) of codi-
4
Figure 2.1. Two families.
V
mension 4 linear subspaces in P( 2 V ). We will sometimes restrict ourselves to open
V2
V2
subsets U ∼
= Spec(k[aij ]) ⊂ G(4, V ) corresponding to linear subspaces of P( V )
P
given by 4 equations Zik − j6=i1 ,i2 ,i3 ,i4 akj · Zj , 1 ≤ k ≤ 4.
Let us denote Πk = V+ (Zik −
P
j6=i1 ,i2 ,i3 ,i4
V
akj · Zj ) ⊂ U × P( 2 V ), 1 ≤ k ≤ 4.
V
Let S be the tautological rank 6 vector bundle over G(4, 2 V ). Then P(S) ,→
V
V
V
G(4, 2 V ) × P( 2 V ) is the universal family over G(4, 2 V ) and P(S)|U = Π1 ∩ Π2 ∩
V
Π3 ∩ Π4 ⊂ U × P( 2 V ).
Consider the trivial family G(4,
V2
V ) × G(2, V ) ,→ G(4,
Lemma 1. The intersection P(S) ∩ (G(4,
V
V
subvariety of G(4, 2 V ) × P( 2 V ).
V2
V2
V
V ) × P( 2 V ).
V ) × G(2, V )) is a smooth connected
V
Proof: It is sufficient to work over U ⊂ G(4, 2 V ) as above. We may assume
V
that Z9 = 1 by restricting to D+ (Z9 ) ⊂ U × P( 2 V ). Without loss of generality we
may assume that 9 6= i1 , i2 , i3 , i4 .
Then P(S)|U ∩D+ (Z9 ) is given inside of
D+ (Z9 ) ∼
= Spec(k[aij ][Z0 , Z1 , Z2 , Z3 , Z4 , Z5 , Z6 , Z7 , Z8 ])
5
by equations
X
ak9 +
akj · Zj − Zik , 1 ≤ k ≤ 4,
j6=i1 ,i2 ,i3 ,i4 ,9
while U × G(2, V ) ∩ D+ (Z9 ) is given by equations
Z0 = (Z2 Z6 − Z3 Z5 ),
Z1 = (Z2 Z8 − Z3 Z7 ),
Z4 = (Z5 Z8 − Z6 Z7 ).
QED
V
V
Let us denote by i : P(S) ,→ G(4, 2 V ) × P( 2 V ) the closed embedding of the
V
universal family π : P(S) → G(4, 2 V ) over the Grassmannian.
Lemma 2. The morphism of coherent sheaves i∗ IP → i∗ OG(4,V2 V )×P(V2 V ) ∼
= OP(S)
is injective.
Proof: We use notation from the proof of Lemma 1. As in Lemma 1, we work in
V
D+ (Z9 ) over U ⊂ G(4, 2 V ).
Let T = Π1 ∩ ... ∩ Πi ⊂ D+ (Z9 ) ⊂ U × P9 and H = T ∩ Πi+1 , 0 ≤ i ≤ 3. Then
H ⊂ T is a Cartier divisor and the corresponding invertible sheaf is OT (1). We will
prove Lemma 2 by induction on i.
If we multiply the short exact sequence
0 → OT (−1) → OT → OT ∩H → 0
by ⊗OT IP , then we get a short exact sequence
0 → IP (−1)|T → IP |T → IP |T ∩H → 0.
The injectivity of the morphism IP (−1)|T → IP |T follows from the induction assumption that IP |T → OT is injective.
Then a diagram chase shows that IP |T ∩H → OT ∩H is injective as well. QED
6
Lemma 3. R1 π∗ (i∗ IP (2)) = 0.
Proof: As in Lemma 1, we work over U ⊂ G(4,
V2
V ). Then it is sufficient to
show that H 1 (P(S)|U , i∗ IP (2)) = 0.
We use notation from the proof of Lemma 2 and do induction on i ∈ {0, 1, 2, 3}.
For any d ∈ Z we have the short exact sequence of coherent sheaves on T
0 → IP |T (d − 1) → IP |T (d) → IP |T ∩H (d) → 0.
Hence vanishing of H i (T ∩ H, IP |T ∩H (3 − i)) follows from vanishing of
H i (T, IP |T (3 − i))
and H i+1 (T, IP |T (3 − i − 1)).
By the cohomological flat base change theorem
H i (U × P9 , i∗ IP (d)) ∼
= k[aij ]⊗k H i (P9 , IP (d)).
So, it is enough to check that for the Plücker ideal IP as a coherent sheaf on P9
we have H i (P9 , IP (3 − i)) = 0 for 1 ≤ i ≤ 5.
V
Let P = G(2, V ) ,→ P( 2 V ) ∼
= P9 be the Plücker embedding. The short exact
sequence
0 → IP (d) → OP9 (d) → OP (d) → 0
and projective normality of the Plücker embedding imply that H 1 (P9 , IP (d)) = 0
and H i (P9 , IP (d)) ∼
= H i−1 (P, OP (d)), 2 ≤ i ≤ 8 for any d ∈ Z.
Finally, H i (P, OP (2 − i)) ∼
= H i (P, ωP (7 − i)) = 0, 1 ≤ i ≤ 4 by the Kodaira
Vanishing theorem. QED
7
Definition 1. E = π∗ (O(2)/i∗ IP (2)).
Corollary 1. There exists a surjective morphism of coherent sheaves π∗ (O(2)) ∼
=
Sym2 (S ∨ ) → E with kernel π∗ (i∗ IP (2)).
Let us describe this kernel more explicitely over an open subset U ⊂ G(4,
V2
V)
as above.
Lemma 4. Over U ⊂ G(4,
V2
V ) one can describe π∗ (i∗ IP (2))|U as a subsheaf of
π∗ (O(2))|U ∼
= Sym2 (S ∨ )|U =
M
OU · Zi Zj
0≤i≤j≤9, i,j6=i1 ,i2 ,i3 ,i4
as follows:
∗
π∗ (i IP (2))|U =
5
X
OU · pi ⊂ π∗ (O(2))|U .
i=1
Proof: We will be working in the graded ring
B = k[aij ][Z0 , ..., Zˆi1 , ..., Zˆi2 , ..., Zˆi3 , ..., Zˆi4 , ..., Z9 ].
Note that P(S)|U ∼
= P roj(B) ⊂ U × P9 .
Let ξ ∈ H 0 (P(S)|U , i∗ IP (2)). Then
ξ|D+ (Zi )
5
X
ωij
· pj
=
(Zi )N
j=1
for some ωij ∈ BN homogeneous elements of B of degree N for any i. Comparing
ξ|D+ (Zi ) and ξ|D+ (Zi0 ) we get
5
X
j=1
ωij · pj = (Zi )N · γi
8
in B for some γi ∈ B2 .
Since i∗ IP ⊂ B is a prime ideal by Lemma 1, we conclude that γi ∈ i∗ IP .
Hence ξ|D+ (Zi ) =
P5
j=1
γj · pj for some γj ∈ k[aij ] for any i. QED
Remark 1. It follows from [14] that E is a locally free sheaf of rank 16 over the
V
open subset of G(4, 2 V ), which is the complement to the union of the following
V
G-orbits (we use the numeration of G-orbits in G(4, 2 V ) from [14]):
• orbits 29, 32, 36, over which the dimension of the fibers of E is 17,
• orbits 33, 35, 37, over which the dimension of the fibers of E is 18,
• orbit 38, over which the dimension of the fibers of E is 20.
As a compactification of a parameter space of complete intersections
G(2, V ) ∩ ( codim 5 quadric in P(
^2
V ))
we take the following space X.
Definition 2. X = P(E).
Let us denote by p : X = P(E) → G(4,
V2
V ) the canonical projection. Note that
G acts on X and p is G-equivariant.
Corollary 2. The action of G on X extends to an action of G on the tautological
line bundle O(1) on X.
Proof: The Plücker ideal IP ⊂ OP(V2 V ) is G-invariant. Hence G acts on E. Hence
it acts on the space of lines in its fibers which is the tautological line bundle O(1).
QED
We will study the following linearizations of G = SL(5)-action on X.
9
Definition 3. La,b = OP(E) (a)⊗O p∗ OG(4,V2 V ) (b), a, b ∈ Z, a, b > 0.
We will also use notation La,b = Lt , where t = b/a. In this thesis we will focus
on the ’asymptotic’ case, when t → ∞.
Chapter
3
Hilbert-Mumford criterion
In this chapter we give a solution to the Hilbert-Mumford numertical criterion of
stability for the GIT problem described in Chapter 2.
Let λ : Gm → G = SL(5) be a 1-PS. We can assume that we have chosen coordinates X1 , X2 , X3 , X4 , X5 in such a way that λ diagonalizes with weights
w1 , w2 , w3 , w4 , w5
such that w1 ≥ w2 ≥ w3 ≥ w4 ≥ w5 and w1 + w2 + w3 + w4 + w5 = 0. Let us denote
by
τ0 = w1 + w2 ,
τ1 = w1 + w3 ,
τ2 = w1 + w4 ,
τ3 = w1 + w5 ,
τ4 = w2 + w3 ,
τ5 = w2 + w4 ,
τ6 = w2 + w5 ,
τ7 = w3 + w4 ,
τ8 = w3 + w5 ,
τ9 = w4 + w5 .
V
the corresponding weights of the coordinates Zi on P( 2 V ) ∼
= P9 .
It follows from Lemma 4 that a point ξ ∈ X is a quadric q in the 6-dimensional
V
projective space π = p(ξ) ∈ G(4, 2 V ) taken modulo Plücker quadrics
p1 = Z0 Z7 − Z1 Z5 + Z2 Z4 , p2 = Z0 Z8 − Z1 Z6 + Z3 Z4 , p3 = Z0 Z9 − Z2 Z6 + Z3 Z5 ,
p4 = Z1 Z9 − Z2 Z8 + Z3 Z7 ,
p5 = Z4 Z9 − Z5 Z8 + Z6 Z7 .
11
We will write ξ = (π, q). Then a standard computation (see [13] or [5]) shows
that the Hilbert-Mumford weight function of ξ can be expressed as follows:
µ(ξ, λ) = a · (τi + τj ) + b · (τi1 + τi2 + τi3 + τi4 ),
where τi1 +τi2 +τi3 +τi4 is the maximal λ-weight at π of the action of G on G(4,
V2
V)
and τi + τj is the maximal λ-weight of the quadratic polynomial q (we take its representative modulo p1 , p2 , p3 , p4 , p5 with the minimal such weight).
Then the Hilbert-Mumford numerical criterion of stability can be stated as follows
([13]):
ξ is semistable if and only if µ(ξ, λ) ≥ 0 for any λ,
(3.1)
ξ is properly stable if and only if µ(ξ, λ) > 0 for any λ,
(3.2)
ξ is unstable if and only if µ(ξ, λ) < 0 for some λ.
(3.3)
Lemma 5. Given a maximal torus in G, there exists a finite set of 1-PS diagonalized by this maximal torus for which it is sufficient to check inequalities (3.1) − (3.3).
Proof: Let us fix ξ = (π, q) ∈ X. Let us denote by τ the ordering induced on τi
by a given 1-PS λ with weights w1 , w2 , w3 , w4 , w5 . Let
τi0 ≥ τi1 ≥ τi2 ≥ τi3 ≥ τi4 ≥ τi5 ≥ τi6 ≥ τi7 ≥ τi8 ≥ τi9 .
Let us denote by τ 2 the ordering induced on τI = τi + τj , i ≤ j, I = {i, j}. We
will choose indices in such a way that τIi+1 ≥ τIi ≥ τI0 for any i ≥ 0.
The choice of coordinates Xi in V determines the flag 0 = E0 ⊂ E1 ⊂ E2 ⊂
E3 ⊂ E4 ⊂ E5 = V , where Ei ⊂ V is the linear subspace cut out by equations
V
Xi+1 = ... = X5 = 0. The ordering τ determines a flag in P( 2 V ). The ordering τ 2
determines a flag in the vector space H 0 (Pπ , O(2)) of quadrics in the 5-dimensional
V
V2
linear subspace Pπ ⊂ P( 2 V ) ∼
= P9 cut out by the equations π ∈ G(4, V ).
Let us notice (see [13] (section 4 in chapter 4) or [5] (chapter 11)) that one can
12
write µ(ξ, λ) in the form
X
X
µ(ξ, λ) = a · (
fi · (τIi+1 − τIi ) + f0 · τI0 ) + b · (
dj · (τij − τij+1 ) + d9 · τ9 ),
i≥0
j≥0
where fi ≥ 0 are the dimensions of intersections of the line determined by the quadric
q with the vector spaces of the flag in H 0 (Pπ , O(2)) defined above and dj ≥ 0 are
the dimensions of intersections of the 4-dimensional space determined by π with the
V
vector spaces of the flag in P( 2 V ) defined above.
Orderings τ and τ 2 determine a convext polytope in the space of wj . Since τi and
τI are linear in wj we can express them as nonnegative linear combinations of their
values at wj corresponding to vertices of this convext polytope. Hence the same
holds for µ(ξ, λ): it is equal to a linear combination with nonnegative coefficients
of µ(ξ, λk ), where λk are 1-PS with the same diagonalizing maximal torus in G,
but whose weights correspond to the (finitely many) vertices of the convex polytope
above. This proves the claim. QED
Let us compute these ’extremal’ destabilizing 1-PS explicitly. It is sufficient to
find all the vertices of convex polytopes whose sides are given by linear equations (in
coordinates wi ) of the form τi = τj and τi + τj = τk + τl for various indices i, j, k, l.
Let us switch to more convenient coordinates first:
c1 =
w1 − w2
,
5
c2 =
w2 − w3
,
5
c3 =
w3 − w4
,
5
c4 =
w4 − w5
.
5
Then ci ≥ 0 for any i and
w1 = 4c1 + 3c2 + 2c3 + c4 ,
w2 = −c1 + 3c2 + 2c3 + c4 ,
w4 = −c1 − 2c2 − 3c3 + c4 ,
w3 = −c1 − 2c2 + 2c3 + c4 ,
w5 = −c1 − 2c2 − 3c3 − 4c4 ,
τ0 = 3c1 + 6c2 + 4c3 + 2c4 , τ1 = 3c1 + c2 + 4c3 + 2c4 , τ2 = 3c1 + c2 − c3 + 2c4 ,
τ3 = 3c1 + c2 − c3 − 3c4 , τ4 = −2c1 + c2 + 4c3 + 2c4 , τ5 = −2c1 + c2 − c3 + 2c4 ,
τ6 = −2c1 + c2 − c3 − 3c4 , τ7 = −2c1 − 4c2 − c3 + 2c4 , τ8 = −2c1 − 4c2 − c3 − 3c4 ,
τ9 = −2c1 − 4c2 − 6c3 − 3c4 .
13
Note that we can talk either about convex polytopes in the projective space of
ci ’s or about convex cones in the affine space of ci ’s.
All possible equations of the form τi = τj and τi + τj = τk + τl can be written in
terms of ci as follows (we list the 4-tuples of coefficients (u1 , u2 , u3 , u4 ) of equations
u1 · c1 + u2 · c2 + u3 · c3 + u4 · c4 = 0):
(1, −1, 0, 0) (1, 0, −1, 0) (1, 0, 0, −1) (0, 1, −1, 0) (0, 1, 0, −1)
(0, 0, 1, −1) (2, 0, 0, −1) (1, 0, 0, −2) (2, 0, −1, 0) (1, 0, −2, 0)
(0, 2, 0, −1) (0, 1, 0, −2) (0, 1, 1, −1) (0, 1, −1, −1) (1, −1, −2, 0)
(2, 1, −1, 0) (1, −1, 1, 0) (1, 0, −2, −2) (2, 0, −1, −1) (1, 2, 0, −2)
(1, −1, 0, 2) (1, −1, 0, −2) (2, 1, 0, −2) (1, 0, −1, −2) (1, 0, 1, −1)
(1, 0, −1, 1) (1, 2, 0, −1) (1, 1, 0, −1) (1, 0, −1, −1) (1, 0, −2, −1)
(1, −1, 0, 1) (1, −1, 0, −1) (2, 1, 0, −1) (2, 0, −1, −2) (2, 0, 1, −1)
(2, 0, −1, 1) (2, 0, −2, −1) (1, 1, 0, −2) (2, 2, 0, −1) (0, 1, −1, 1)
(0, 1, −1, −2) (0, 2, 1, −1) (1, 1, −1, 0) (1, −1, −1, 0) (1, −1, 1, 2)
(2, 1, −1, 1) (1, −1, 1, −1) (2, 1, −1, −2) (1, −1, −1, −1) (1, −1, −3, −1)
(2, 3, 1, −1) (1, −1, −3, −2) (1, 2, 1, −1) (1, −1, −2, −1) (1, 1, 1, −1)
(1, 3, 1, −1) (1, 1, −1, −1) (1, −1, −2, −2) (2, 2, 1, −1) (2, 1, −1, −1)
(1, 1, −1, −2) (1, −1, 1, 1) (1, 1, −1, 1) (1, −1, −1, −2) (2, 1, 1, −1)
(1, −1, −1, 1) (1, 0, 0, 0) (0, 1, 0, 0) (0, 0, 1, 0) (0, 0, 0, 1).
Now one has to go over all triples of these equations and whenever their common
solution is a line in the first hyperoctant of the space of ci , this solution will be a
vertex of a convex polytope (or an extremal ray of a convex cone) as above. We
used an Excel macros (see Appendix C) to do this computation and obtained the
following list of vertices (we list the representative nonzero 4-tuples (c1 , c2 , c3 , c4 )):
(1, 1, 1, 1) (1, 1, 1, 2) (2, 2, 2, 1) (0, 0, 0, 1) (2, 2, 2, 3) (1, 1, 1, 3) (1, 1, 1, 4)
(1, 1, 1, 5) (1, 1, 1, 6) (1, 1, 2, 1) (2, 2, 1, 2) (1, 1, 0, 1) (1, 1, 3, 1) (1, 1, 4, 1)
(1, 1, 2, 2) (2, 2, 1, 1) (0, 0, 1, 1) (1, 1, 3, 3) (4, 4, 1, 1) (2, 2, 3, 3) (3, 3, 1, 1)
14
(3, 3, 2, 2) (1, 1, 4, 4) (2, 2, 1, 4) (1, 1, 0, 2) (1, 1, 3, 2) (1, 1, 4, 2) (1, 1, 5, 2)
(2, 2, 4, 1) (2, 2, 0, 1) (2, 2, 6, 1) (2, 2, 3, 1) (4, 4, 1, 2) (2, 2, 5, 1) (4, 4, 3, 2)
(2, 2, 7, 1) (1, 1, 2, 3) (2, 2, 4, 3) (1, 1, 2, 4) (1, 1, 2, 6) (1, 1, 2, 7) (1, 1, 2, 5)
(2, 2, 1, 3) (4, 4, 2, 1) (2, 2, 1, 6) (4, 4, 2, 3) (2, 2, 1, 5) (2, 2, 1, 8) (4, 4, 2, 5)
(2, 2, 1, 9) (2, 2, 1, 11) (2, 2, 1, 7) (1, 1, 3, 4) (3, 3, 1, 4) (2, 2, 0, 3) (1, 1, 0, 3)
(1, 1, 0, 4) (1, 1, 0, 5) (2, 2, 6, 3) (1, 1, 3, 5) (1, 1, 3, 7) (1, 1, 3, 8) (1, 1, 3, 6)
(2, 2, 5, 3) (2, 2, 7, 3) (4, 4, 1, 6) (2, 2, 9, 3) (1, 1, 4, 3) (3, 3, 4, 1) (1, 1, 5, 4)
(3, 3, 5, 2) (1, 1, 5, 3) (1, 1, 6, 3) (1, 1, 6, 4) (3, 3, 7, 1) (1, 1, 7, 4) (2, 1, 2, 2)
(1, 2, 1, 1) (1, 0, 1, 1) (1, 3, 1, 1) (1, 4, 1, 1) (1, 2, 1, 2) (2, 1, 2, 1) (0, 1, 0, 1)
(3, 1, 3, 1) (1, 3, 1, 3) (1, 4, 1, 2) (1, 3, 1, 2) (2, 3, 2, 4) (1, 5, 1, 2) (2, 1, 2, 4)
(1, 0, 1, 2) (1, 6, 1, 2) (4, 1, 4, 2) (2, 3, 2, 1) (2, 4, 2, 1) (2, 0, 2, 1) (2, 6, 2, 1)
(2, 5, 2, 1) (0, 1, 0, 2) (0, 2, 0, 1) (0, 1, 0, 3) (1, 2, 1, 4) (5, 1, 5, 2) (3, 2, 3, 4)
(3, 1, 3, 2) (2, 3, 2, 6) (1, 3, 1, 6) (4, 2, 4, 1) (3, 2, 3, 1) (1, 6, 1, 3) (3, 4, 3, 2)
(1, 2, 1, 3) (2, 1, 2, 3) (1, 4, 1, 3) (2, 4, 2, 5) (1, 2, 1, 5) (2, 4, 2, 3) (1, 2, 1, 6)
(1, 2, 1, 8) (1, 2, 1, 9) (1, 2, 1, 7) (4, 1, 4, 3) (2, 5, 2, 6) (4, 3, 4, 5) (1, 7, 1, 3)
(3, 5, 3, 1) (1, 6, 1, 4) (3, 1, 3, 5) (1, 5, 1, 3) (2, 1, 2, 6) (1, 0, 1, 3) (1, 8, 1, 3)
(3, 1, 3, 9) (2, 1, 2, 7) (1, 2, 2, 1) (2, 1, 1, 2) (3, 1, 1, 3) (1, 3, 3, 1) (1, 0, 0, 1)
(2, 1, 4, 2) (1, 3, 2, 1) (1, 0, 2, 1) (1, 4, 2, 1) (1, 5, 2, 1) (2, 4, 1, 2) (2, 3, 1, 2)
(2, 0, 1, 2) (2, 6, 1, 2) (2, 5, 1, 2) (4, 1, 2, 4) (2, 7, 1, 2) (4, 2, 1, 4) (2, 1, 5, 2)
(2, 1, 0, 2) (2, 1, 6, 2) (2, 1, 3, 2) (2, 1, 7, 2) (1, 2, 4, 1) (1, 2, 0, 1) (1, 2, 3, 1)
(1, 2, 5, 1) (1, 4, 3, 1) (1, 3, 5, 1) (1, 3, 0, 1) (1, 3, 4, 1) (1, 3, 6, 1) (1, 0, 3, 1)
(1, 5, 3, 1) (1, 6, 3, 1) (1, 2, 2, 2) (2, 1, 1, 1) (3, 1, 1, 1) (0, 1, 1, 1) (4, 1, 1, 1)
15
(3, 2, 2, 2) (5, 1, 1, 1) (6, 1, 1, 1) (2, 1, 1, 4) (1, 4, 4, 2) (3, 1, 1, 6) (1, 0, 0, 2)
(2, 3, 3, 4) (1, 5, 5, 2) (1, 3, 3, 2) (3, 2, 2, 6) (4, 1, 1, 8) (2, 4, 4, 1) (4, 1, 1, 2)
(6, 2, 2, 3) (2, 0, 0, 1) (2, 3, 3, 1) (2, 5, 5, 1) (4, 3, 3, 2) (6, 1, 1, 3) (8, 1, 1, 4)
(1, 2, 2, 4) (2, 4, 4, 5) (1, 2, 2, 3) (1, 2, 2, 5) (2, 4, 4, 3) (1, 2, 2, 6) (1, 2, 2, 9)
(1, 2, 2, 10) (1, 2, 2, 7) (1, 2, 2, 8) (4, 2, 2, 1) (2, 1, 1, 3) (4, 2, 2, 5) (2, 1, 1, 5)
(4, 2, 2, 3) (2, 1, 1, 6) (2, 1, 1, 8) (2, 1, 1, 7) (3, 1, 1, 2) (0, 1, 1, 2) (6, 1, 1, 2)
(3, 2, 2, 4) (5, 1, 1, 2) (5, 2, 2, 4) (8, 1, 1, 2) (7, 1, 1, 2) (6, 2, 2, 1) (0, 2, 2, 1)
(3, 4, 4, 2) (3, 2, 2, 1) (5, 2, 2, 1) (5, 4, 4, 2) (9, 2, 2, 1) (10, 2, 2, 1) (7, 2, 2, 1)
(8, 2, 2, 1) (3, 1, 1, 5) (6, 2, 2, 5) (6, 2, 2, 7) (3, 1, 1, 4) (3, 1, 1, 7) (3, 1, 1, 8)
(3, 1, 1, 10) (3, 1, 1, 9) (0, 1, 1, 3) (0, 1, 1, 4) (8, 1, 1, 3) (4, 3, 3, 5) (3, 5, 5, 1)
(5, 1, 1, 9) (4, 1, 1, 7) (4, 1, 1, 3) (6, 1, 1, 4) (10, 1, 1, 6) (8, 1, 1, 5) (4, 6, 6, 1)
(7, 1, 1, 3) (5, 3, 3, 1) (1, 4, 4, 3) (1, 6, 6, 4) (5, 2, 2, 6) (3, 4, 4, 5) (3, 4, 4, 1)
(4, 1, 1, 6) (5, 1, 1, 3) (1, 3, 3, 5) (7, 2, 2, 6) (5, 4, 4, 3) (1, 5, 5, 3) (5, 3, 3, 4)
(6, 1, 1, 10) (5, 1, 1, 8) (9, 1, 1, 5) (7, 1, 1, 4) (10, 1, 1, 3) (9, 1, 1, 3) (2, 4, 1, 4)
(1, 2, 0, 2) (1, 2, 4, 2) (1, 2, 3, 2) (1, 2, 6, 2) (1, 2, 5, 2) (2, 1, 4, 1) (2, 1, 0, 1)
(4, 2, 1, 2) (2, 1, 5, 1) (2, 1, 3, 1) (4, 2, 3, 2) (2, 1, 6, 1) (1, 3, 2, 3) (3, 1, 6, 1)
(1, 4, 2, 4) (2, 3, 4, 3) (2, 3, 1, 3) (6, 2, 3, 2) (4, 1, 2, 1) (4, 3, 2, 3) (2, 5, 1, 5)
(6, 1, 3, 1) (3, 1, 0, 1) (5, 1, 2, 1) (5, 3, 1, 3) (3, 1, 7, 1) (1, 3, 5, 3) (3, 4, 1, 4)
(6, 2, 1, 2) (6, 1, 2, 1) (3, 1, 5, 1) (3, 2, 4, 2) (0, 1, 2, 1) (3, 1, 4, 1) (3, 1, 2, 1)
(6, 2, 5, 2) (3, 1, 8, 1) (9, 3, 1, 3) (3, 2, 1, 2) (4, 1, 6, 1) (5, 2, 4, 2) (5, 4, 3, 4)
(8, 1, 2, 1) (9, 1, 2, 1) (7, 1, 2, 1) (7, 2, 1, 2) (1, 4, 2, 2) (1, 0, 2, 2) (1, 3, 2, 2)
(2, 3, 4, 4) (1, 5, 2, 2) (2, 1, 4, 4) (1, 6, 2, 2) (1, 7, 2, 2) (4, 1, 2, 2) (2, 0, 1, 1)
(2, 3, 1, 1) (2, 4, 1, 1) (2, 5, 1, 1) (5, 1, 2, 2) (1, 2, 4, 4) (8, 1, 2, 2) (3, 2, 4, 4)
16
(6, 1, 2, 2) (0, 1, 2, 2) (3, 1, 2, 2) (5, 2, 4, 4) (9, 1, 2, 2) (11, 1, 2, 2) (7, 1, 2, 2)
(4, 2, 1, 1) (0, 2, 1, 1) (3, 2, 1, 1) (3, 4, 2, 2) (6, 2, 1, 1) (7, 2, 1, 1) (5, 2, 1, 1)
(4, 0, 1, 1) (3, 0, 1, 1) (3, 0, 2, 2) (5, 0, 1, 1) (5, 3, 1, 1) (4, 5, 1, 1) (1, 4, 3, 3)
(3, 4, 1, 1) (3, 5, 2, 2) (2, 5, 3, 3) (4, 6, 1, 1) (4, 3, 1, 1) (4, 7, 1, 1) (4, 1, 3, 3)
(6, 1, 4, 4) (3, 5, 1, 1) (3, 7, 2, 2) (1, 7, 3, 3) (3, 6, 1, 1) (0, 3, 1, 1) (3, 6, 2, 2)
(3, 9, 2, 2) (7, 3, 1, 1) (8, 3, 1, 1) (6, 3, 1, 1) (2, 8, 1, 4) (2, 3, 1, 4) (2, 5, 1, 4)
(2, 0, 1, 4) (2, 10, 1, 4) (2, 6, 1, 4) (2, 9, 1, 4) (4, 3, 2, 8) (2, 11, 1, 4) (6, 1, 3, 12)
(4, 1, 2, 8) (2, 7, 1, 4) (1, 4, 6, 2) (1, 4, 3, 2) (1, 4, 0, 2) (1, 4, 5, 2) (1, 4, 8, 2)
(1, 4, 7, 2) (2, 1, 3, 4) (1, 5, 3, 2) (1, 6, 4, 2) (4, 2, 1, 8) (5, 1, 2, 10) (2, 3, 7, 4)
(1, 5, 7, 2) (2, 1, 5, 4) (1, 3, 5, 2) (1, 5, 4, 2) (2, 3, 0, 4) (1, 5, 0, 2) (2, 1, 0, 4)
(1, 3, 0, 2) (3, 1, 0, 6) (2, 3, 6, 4) (2, 3, 8, 4) (2, 3, 5, 4) (2, 3, 11, 4) (2, 3, 9, 4)
(1, 5, 6, 2) (1, 5, 9, 2) (1, 5, 8, 2) (2, 1, 6, 4) (1, 0, 3, 2) (1, 7, 3, 2) (1, 8, 3, 2)
(1, 6, 3, 2) (2, 1, 8, 4) (2, 1, 9, 4) (2, 1, 7, 4) (1, 3, 4, 2) (1, 3, 7, 2) (1, 3, 6, 2)
(1, 0, 4, 2) (1, 8, 4, 2) (1, 9, 4, 2) (1, 7, 4, 2) (3, 1, 4, 6) (4, 1, 8, 2) (2, 5, 4, 1)
(2, 0, 4, 1) (2, 6, 4, 1) (2, 3, 4, 1) (2, 8, 4, 1) (2, 7, 4, 1) (8, 2, 3, 4) (4, 1, 9, 2)
(4, 1, 0, 2) (4, 1, 6, 2) (4, 1, 10, 2) (4, 1, 3, 2) (4, 1, 5, 2) (4, 1, 11, 2) (12, 3, 1, 6)
(8, 2, 1, 4) (4, 1, 7, 2) (6, 4, 1, 3) (2, 4, 3, 1) (4, 3, 1, 2) (2, 6, 5, 1) (4, 5, 3, 2)
(2, 4, 8, 1) (2, 3, 7, 1) (2, 5, 3, 1) (4, 5, 1, 2) (2, 7, 5, 1) (4, 7, 3, 2) (2, 4, 0, 1)
(2, 3, 0, 1) (2, 0, 3, 1) (2, 7, 3, 1) (2, 6, 3, 1) (4, 0, 1, 2) (2, 0, 5, 1) (4, 0, 3, 2)
(6, 0, 1, 3) (4, 8, 1, 2) (2, 4, 5, 1) (4, 8, 3, 2) (2, 4, 6, 1) (2, 4, 9, 1) (2, 4, 7, 1)
(4, 6, 1, 2) (8, 1, 2, 4) (4, 9, 1, 2) (4, 7, 1, 2) (2, 3, 5, 1) (4, 6, 3, 2) (2, 3, 8, 1)
(2, 3, 6, 1) (2, 9, 5, 1) (2, 8, 5, 1) (4, 11, 3, 2) (4, 9, 3, 2) (10, 2, 1, 5) (1, 3, 2, 6)
17
(5, 1, 10, 2) (3, 2, 6, 4) (2, 3, 4, 6) (3, 1, 6, 2) (1, 4, 2, 8) (2, 6, 4, 3) (4, 2, 8, 1)
(1, 6, 2, 3) (3, 2, 6, 1) (1, 8, 2, 4) (3, 4, 6, 2) (1, 3, 2, 5) (2, 1, 4, 5) (1, 6, 2, 4)
(1, 5, 2, 3) (1, 0, 2, 3) (1, 0, 2, 4) (2, 6, 4, 7) (2, 6, 4, 5) (1, 3, 2, 7) (1, 3, 2, 4)
(1, 3, 2, 8) (1, 3, 2, 12) (1, 3, 2, 13) (1, 3, 2, 9) (1, 3, 2, 10) (2, 5, 4, 6) (4, 1, 8, 3)
(2, 7, 4, 8) (4, 3, 8, 5) (4, 5, 8, 7) (1, 7, 2, 3) (1, 9, 2, 4) (3, 5, 6, 1) (3, 7, 6, 2)
(1, 4, 2, 3) (3, 4, 6, 5) (1, 8, 2, 5) (2, 1, 4, 6) (1, 8, 2, 3) (1, 9, 2, 3) (2, 3, 4, 8)
(3, 1, 6, 5) (3, 2, 6, 7) (2, 1, 4, 3) (1, 5, 2, 4) (1, 7, 2, 4) (1, 10, 2, 4) (1, 11, 2, 4)
(3, 1, 6, 12) (2, 1, 4, 8) (3, 5, 6, 4) (3, 1, 6, 8) (3, 8, 6, 1) (2, 1, 4, 9) (2, 3, 1, 6)
(4, 3, 2, 6) (10, 2, 5, 4) (6, 4, 3, 8) (8, 1, 4, 2) (6, 2, 3, 4) (2, 4, 1, 8) (8, 3, 4, 6)
(4, 5, 2, 10) (6, 1, 3, 2) (10, 1, 5, 2) (2, 5, 1, 10) (4, 6, 2, 3) (2, 6, 1, 3) (6, 4, 3, 2)
(2, 10, 1, 5) (6, 2, 3, 1) (8, 2, 4, 1) (4, 1, 2, 3) (2, 4, 1, 5) (2, 4, 1, 3) (6, 4, 3, 1)
(2, 6, 1, 5) (4, 3, 2, 1) (4, 5, 2, 3) (2, 0, 1, 3) (4, 0, 2, 1) (4, 0, 2, 3) (2, 0, 1, 5)
(4, 6, 2, 1) (4, 6, 2, 7) (2, 3, 1, 8) (2, 3, 1, 5) (2, 3, 1, 7) (4, 6, 2, 5) (2, 3, 1, 10)
(2, 3, 1, 12) (2, 3, 1, 14) (2, 3, 1, 9) (2, 3, 1, 11) (2, 8, 1, 3) (4, 1, 2, 6) (2, 5, 1, 3)
(2, 7, 1, 3) (2, 9, 1, 3) (4, 10, 2, 3) (2, 12, 1, 5) (6, 8, 3, 1) (8, 6, 4, 1) (8, 2, 4, 3)
(2, 8, 1, 6) (6, 2, 3, 7) (2, 10, 1, 7) (4, 5, 2, 1) (4, 8, 2, 1) (4, 7, 2, 1) (4, 3, 2, 10)
(6, 1, 3, 8) (4, 1, 2, 5) (4, 7, 2, 3) (2, 7, 1, 5) (6, 5, 3, 1) (6, 1, 3, 5) (4, 1, 2, 9)
(4, 8, 2, 3) (8, 1, 4, 6) (4, 12, 2, 3) (4, 9, 2, 3) (2, 8, 1, 5) (4, 1, 2, 10) (2, 11, 1, 5)
(2, 13, 1, 5) (6, 2, 3, 15) (2, 9, 1, 5) (6, 4, 3, 5) (6, 7, 3, 2) (2, 4, 1, 9) (6, 5, 3, 4)
(2, 9, 1, 6) (6, 4, 1, 8) (7, 1, 3, 2) (5, 3, 1, 6) (7, 3, 2, 6) (9, 1, 4, 2) (5, 1, 11, 2)
(3, 2, 8, 4) (1, 3, 5, 6) (3, 1, 7, 2) (1, 4, 6, 8) (2, 3, 7, 6) (10, 2, 1, 4) (10, 1, 3, 2)
(5, 1, 8, 2) (3, 1, 4, 2) (5, 2, 6, 4) (5, 3, 4, 6) (5, 1, 7, 2) (5, 1, 3, 2) (10, 2, 3, 4)
(5, 1, 6, 2) (5, 1, 12, 2) (5, 1, 4, 2) (5, 1, 0, 2) (5, 1, 13, 2) (15, 3, 2, 6) (5, 1, 9, 2)
18
(3, 2, 1, 4) (3, 2, 7, 4) (3, 2, 10, 4) (3, 2, 0, 4) (3, 2, 5, 4) (3, 2, 12, 4) (3, 2, 9, 4)
(9, 2, 1, 4) (1, 2, 3, 4) (3, 1, 5, 2) (1, 2, 5, 4) (6, 1, 8, 2) (0, 1, 3, 2) (5, 2, 1, 4)
(6, 2, 1, 4) (8, 1, 3, 2) (8, 3, 1, 6) (3, 1, 8, 2) (3, 1, 0, 2) (3, 1, 9, 2) (1, 2, 6, 4)
(1, 2, 0, 4) (1, 2, 8, 4) (1, 2, 7, 4) (7, 2, 6, 4) (7, 1, 10, 2) (1, 3, 4, 6) (3, 4, 2, 8)
(1, 3, 8, 6) (12, 1, 3, 2) (14, 1, 3, 2) (9, 1, 3, 2) (11, 1, 3, 2) (4, 3, 5, 6) (6, 1, 9, 2)
(6, 4, 1, 2) (8, 2, 3, 1) (8, 6, 1, 3) (4, 2, 10, 1) (3, 2, 8, 1) (1, 6, 8, 3) (3, 4, 1, 2)
(5, 6, 1, 3) (1, 6, 5, 3) (5, 8, 3, 4) (4, 2, 7, 1) (3, 2, 5, 1) (4, 6, 5, 3) (0, 2, 3, 1)
(4, 2, 5, 1) (4, 2, 3, 1) (8, 4, 3, 2) (4, 2, 6, 1) (4, 2, 9, 1) (8, 4, 7, 2) (4, 2, 0, 1)
(4, 2, 11, 1) (12, 6, 1, 3) (8, 4, 1, 2) (5, 2, 3, 1) (5, 4, 1, 2) (3, 8, 1, 4) (3, 2, 4, 1)
(7, 2, 3, 1) (7, 6, 2, 3) (3, 2, 7, 1) (6, 4, 5, 2) (3, 2, 0, 1) (3, 2, 9, 1) (5, 4, 6, 2)
(5, 6, 4, 3) (5, 2, 8, 1) (2, 6, 7, 3) (7, 4, 6, 2) (7, 8, 5, 4) (12, 2, 3, 1) (13, 2, 3, 1)
(9, 2, 3, 1) (10, 2, 3, 1) (9, 4, 1, 2) (4, 2, 1, 3) (1, 2, 4, 6) (2, 1, 5, 6) (3, 4, 1, 5)
(4, 5, 1, 6) (8, 2, 1, 3) (4, 2, 3, 5) (7, 1, 2, 3) (1, 2, 3, 5) (5, 2, 1, 3) (6, 3, 1, 4)
(5, 1, 3, 4) (6, 3, 2, 5) (5, 3, 1, 2) (7, 3, 2, 1) (3, 5, 4, 1) (5, 3, 2, 1) (10, 4, 1, 3)
(7, 3, 1, 2) (3, 5, 1, 4) (8, 3, 1, 2) (6, 4, 1, 7) (8, 2, 3, 6) (10, 4, 3, 9) (4, 2, 1, 5)
(8, 2, 3, 9) (7, 5, 1, 3) (5, 1, 2, 7) (5, 3, 1, 11) (7, 1, 3, 9) (5, 1, 2, 6) (5, 1, 2, 4)
(7, 5, 1, 2) (7, 1, 3, 6) (7, 3, 2, 4) (5, 1, 2, 12) (4, 2, 1, 9) (9, 1, 4, 3) (9, 7, 1, 3)
(8, 4, 2, 1) (9, 5, 2, 6) (4, 2, 1, 11) (4, 1, 9, 3) (2, 5, 9, 6) (1, 7, 9, 3) (1, 4, 6, 3)
(1, 2, 4, 5) (2, 1, 5, 7) (1, 2, 4, 3) (1, 3, 5, 7) (3, 1, 7, 13) (2, 1, 5, 9) (1, 5, 7, 3)
(3, 1, 7, 9) (1, 2, 4, 8) (2, 1, 5, 10) (6, 8, 2, 1) (6, 7, 1, 2) (4, 5, 1, 7) (3, 7, 4, 2)
(3, 5, 2, 4) (6, 7, 1, 10) (2, 5, 3, 6) (3, 5, 2, 1) (5, 7, 2, 1) (5, 9, 4, 2) (2, 8, 6, 3)
(1, 7, 6, 3) (1, 6, 5, 4) (1, 8, 7, 5) (5, 6, 1, 8) (5, 6, 1, 2) (3, 7, 4, 1) (1, 5, 4, 3)
19
(2, 7, 5, 3) (1, 4, 3, 6) (5, 7, 2, 4) (1, 7, 6, 4) (2, 9, 7, 3) (5, 7, 2, 6) (3, 4, 1, 9)
(4, 7, 3, 6) (8, 10, 3, 1) (6, 4, 2, 1) (12, 2, 1, 5) (8, 6, 3, 1) (6, 7, 2, 1) (6, 5, 2, 1)
(6, 5, 1, 2) (4, 1, 5, 3) (6, 2, 7, 5) (6, 4, 5, 7) (4, 6, 7, 1) (5, 1, 3, 7) (3, 1, 2, 4)
(3, 4, 5, 1) (3, 5, 4, 2) (4, 2, 5, 3) (3, 7, 5, 1) (6, 5, 2, 8) (4, 3, 1, 5) (2, 4, 3, 5)
(4, 1, 7, 3) (2, 4, 7, 5) (6, 2, 1, 5) (8, 2, 1, 6) (10, 3, 1, 8) (4, 1, 11, 3) (8, 2, 5, 6)
(4, 1, 12, 3) (4, 1, 0, 3) (4, 1, 6, 3) (8, 1, 6, 5) (8, 3, 2, 7) (2, 6, 3, 7) (2, 4, 9, 5)
(2, 6, 11, 7) (10, 1, 7, 6) (6, 1, 5, 4) (5, 7, 3, 1) (3, 11, 1, 4) (1, 9, 3, 4) (4, 6, 3, 1)
(7, 9, 3, 1) (5, 7, 8, 1) (5, 9, 6, 2) (1, 11, 3, 5) (7, 11, 6, 2) (3, 7, 9, 2) (2, 8, 9, 3)
(3, 5, 9, 1) (5, 3, 6, 1) (7, 5, 3, 1) (11, 1, 3, 5) (5, 3, 11, 1) (7, 3, 6, 2) (13, 7, 1, 3)
(9, 5, 1, 2) (9, 1, 2, 4) (7, 1, 9, 3) (13, 1, 2, 6) (1, 4, 5, 3) (1, 4, 7, 3) (1, 4, 9, 3)
(1, 4, 0, 3) (1, 4, 8, 3) (2, 6, 9, 5) (7, 2, 3, 8) (4, 2, 7, 5) (5, 2, 3, 6) (6, 3, 4, 1)
(5, 0, 1, 2) (6, 1, 2, 8) (3, 1, 2, 5) (3, 7, 1, 4) (1, 5, 3, 4) (2, 1, 3, 5) (3, 9, 1, 4)
(6, 3, 2, 8) (3, 12, 1, 4) (3, 0, 1, 4) (3, 6, 1, 4) (1, 2, 5, 6) (6, 3, 1, 7) (4, 2, 3, 7)
(3, 6, 2, 5) (1, 2, 6, 5) (4, 3, 5, 1) (3, 6, 4, 1) (3, 9, 4, 1) (3, 0, 4, 1) (3, 8, 4, 1)
(2, 4, 5, 3) (1, 2, 7, 6) (3, 6, 5, 2) (1, 8, 4, 3) (3, 2, 1, 7) (1, 2, 7, 5) (1, 3, 9, 7)
(8, 1, 3, 10) (4, 1, 3, 6) (1, 2, 5, 3) (1, 3, 6, 4) (3, 4, 2, 1) (5, 6, 2, 1) (5, 4, 2, 1)
(7, 2, 1, 5) (7, 3, 1, 5) (9, 3, 1, 7) (4, 5, 6, 1) (1, 6, 3, 5) (5, 7, 4, 2) (3, 5, 7, 2)
(2, 5, 6, 3) (3, 4, 8, 1) (4, 3, 9, 1) (7, 1, 4, 6) (5, 3, 4, 2) (5, 2, 6, 3) (1, 3, 7, 5)
(2, 1, 6, 5) (3, 1, 5, 7) (2, 1, 0, 5) (5, 1, 4, 3) (5, 3, 2, 4) (9, 3, 2, 8) (3, 1, 2, 8)
(1, 3, 6, 8) (3, 1, 4, 10) (2, 1, 3, 7) (3, 9, 1, 7) (6, 2, 1, 13) (2, 4, 7, 3) (1, 2, 8, 6)
(1, 3, 10, 8) (3, 6, 8, 2) (3, 6, 7, 1) (3, 9, 7, 1) (3, 9, 8, 2) (1, 9, 5, 3) (7, 1, 5, 4)
(7, 5, 4, 6) (6, 3, 5, 2) (6, 9, 5, 2) (9, 3, 4, 10) (2, 1, 7, 6) (6, 3, 7, 4) (10, 5, 1, 2)
(1, 2, 3, 7) (6, 2, 5, 9) (11, 1, 2, 4) (2, 1, 3, 8) (1, 0, 0, 0) (0, 1, 0, 0) (0, 0, 1, 0)
20
(0, 1, 1, 0) (2, 0, 1, 0) (3, 0, 1, 0) (1, 0, 1, 0) (1, 0, 2, 0) (1, 1, 0, 0) (1, 1, 1, 0)
(2, 2, 1, 0) (1, 1, 2, 0) (1, 1, 3, 0) (1, 2, 2, 0) (4, 1, 1, 0) (2, 3, 1, 0) (1, 3, 2, 0)
(1, 2, 1, 0) (3, 1, 1, 0) (2, 1, 1, 0).
Chapter
4
The asymptotic case
In this chapter we consider the case t → ∞, where t = b/a and
Lt = O(1)⊗p∗ OG(4,V2 V ) (t).
In this case, Lemma 5 allows us to ignore the O(1) term. More precisely, we have
Lemma 6. Assume that the action of G on G(4,
V2
V ) has the unique semistable
orbit which is also properly stable. If t → ∞, then the semistable, properly stable and
V
unstable loci on X are the preimages of such loci on G(4, 2 V ). Moreover, there are
no properly semistable points on X.
Proof: Indeed, let us denote by
µ(ξ, λ) = a · µ(q, λ) + b · µ(π, λ)
the decomposition from the beginning of Chapter 3. We may assume that λ is an
element of a finite set of 1-PS listed in Chapter 3. If t → ∞, then µ(π, λ) > 0 implies
µ(ξ, λ) > 0 and µ(π, λ) < 0 implies µ(ξ, λ) < 0.
Because of the assumption made in the Lemma, we can assume that either of
these two inequalities holds. Then we conclude that X does not contain properly
semistable points (when t → ∞) and the Lemma follows. QED
22
Let us check the assumption of Lemma 6.
Lemma 7. The action of G on G(4,
V2
V ) has the unique semistable orbit which
is also properly stable. It corresponds to the codimension 4 linear sections of the
Grassmannian which are smooth del Pezzo surfaces of degree 5.
Proof: We refer the reader to Appendix A which contains the list of all codimension 4 linear sections of the Grassmannian of lines in P4 which coincides with the list
V
of all orbits of G = SL(5) on G(4, 2 V ). The unique open orbit gives the sections
which are smooth del Pezzo surfaces. We will show that this is the only properly
V
stable G-orbit on G(4, 2 V ) and the other orbits are unstable.
Let us list destabilising 1-PS for each of the non-open orbits. In the list below we
number such orbits according to their numbering in Proposition 2.1 in [14] and for
each orbit provide a destabilizing 1-PS in the form (w1 , w2 , w3 , w4 , w5 ). In each case
the diagonalizing maximal torus is the standard maximal torus in SL(5) ∼
= SL(V ),
where the isomorphism is given by the coordinates X1 , X2 , X3 , X4 , X5 used by Ozeki.
orbit 3 : (9, 19, −11, −1, −16); orbit 4 : (0, −1, 2, 1, −2);
orbit 5 : (1, −1, 2, 0, −2); orbit 6 : (−2, 0, −1, 1, 2); orbit 7 : (1, −1, 2, 0, −2);
orbit 8 : (0, −2, 2, 1, −1); orbit 9 : (−2, 0, −1, 1, 2); orbit 10 : (1, 0, −2, −1, 2);
orbit 11 : (2, 1, −1, 0, −2); orbit 12 : (−1, −2, 2, 1, 0); orbit 13 : (−1, −2, 2, 1, 0);
orbit 14 : (0, −2, 2, 1, −1); orbit 15 : (−1, 2, −2, 1, 0); orbit 16 : (−1, −2, 2, 1, 0);
orbit 17 : (−2, 2, −1, 1, 0); orbit 18 : (−1, −2, 2, 1, 0); orbit 19 : (−1, 2, −2, 1, 0);
orbit 20 : (−1, 2, −2, 1, 0); orbit 21 : (−2, 0, 2, −1, 1); orbit 22 : (−1, 2, −2, 1, 0);
orbit 23 : (2, −1, −2, 1, 0); orbit 24 : (0, −1, 2, −2, 1); orbit 25 : (0, −1, 2, 1, −2);
orbit 26 : (−2, 2, 0, −1, 1); orbit 27 : (−1, 1, −2, 2, 0); orbit 28 : (0, −2, 2, 1, −1);
orbit 29 : (−2, 2, 1, 0, −1); orbit 30 : (−1, −2, 2, 1, 0); orbit 31 : (2, 0, −1, −2, 1);
orbit 32 : (2, −2, 1, 0, −1); orbit 33 : (2, −1, −2, 1, 0); orbit 34 : (1, −2, 2, −1, 0);
orbit 35 : (0, −2, 2, 1, −1); orbit 36 : (−2, 2, 1, −1, 0); orbit 37 : (−2, 2, 1, −1, 0);
orbit 2 : (13, 8, −2, −7, −12); orbit 38 : (−1, 2, 1, 0, −2).
23
It remains to show that the open G-orbit in G(4,
V2
V ) is properly stable. Each
of the ’extremal’ destabilizing 1-PS above gives a set of Plücker coordinates which
have a positive weight and hence have to vanish in order to lead to a point which is
not properly stable. We used an Excel macros (see Appendix D) in order to identify
minimal such sets and for each of them we used MATLAB in order to check that the
orbit of the point in the Grassmannian for which these Plücker coordinates vanish is
not general.
Let us describe this process in more detail. In Appendix E we provide the MATLAB code which can be used in order to check that the general elements of the
V
loci in the Grassmannian G(4, 2 V ) listed below in the form of matrices lie in the
complement of the general G-orbit. Together with each locus we provide Command
2 used in the MATLAB code as well as the rank of the matrix A associated in Appendix E to a matrix representing the general element of this locus (as computed by
MATLAB).

0

0

0

0

0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ ∗ ∗ ∗
MATLAB Command 2: c10 = 0, c11 = 0, c12 = 0, c20 = 0, c21 = 0, c22 = 0,
c23 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c34 = 0, c40 = 0, c41 = 0, c42 = 0,
c43 = 0, c44 = 0, c45 = 0
rank = 35,

0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗



0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗


0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗


0 0 0 0 0 0 0 ∗ ∗ ∗
24
MATLAB Command 2: c10 = 0, c11 = 0, c20 = 0, c21 = 0, c22 = 0, c30 = 0,
c31 = 0, c32 = 0, c33 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0,
c46 = 0
rank = 39,

0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗



0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗


0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ 


0 0 0 0 0 0 ∗ ∗ ∗ ∗
MATLAB Command 2: c10 = 0, c11 = 0, c20 = 0, c21 = 0, c22 = 0, c30 = 0,
c31 = 0, c32 = 0, c33 = 0, c34 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0,
c45 = 0
rank = 38,

0

0

0

0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 0 ∗ ∗ ∗
MATLAB Command 2: c10 = 0, c20 = 0, c21 = 0, c30 = 0, c31 = 0, c32 = 0,
c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0
rank = 39,

0

0

0

0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ ∗ ∗ ∗
MATLAB Command 2: c10 = 0, c20 = 0, c21 = 0, c22 = 0, c23 = 0, c30 = 0,
25
c31 = 0, c32 = 0, c33 = 0, c34 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0,
c45 = 0
rank = 35,

∗

0

0

0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ ∗ ∗ ∗

0 0 0 0 0 0 0 ∗ ∗
MATLAB Command 2: c20 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c34 = 0,
c35 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0, c47 = 0
rank = 39,

∗

0

0

0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 0 0 ∗ ∗
MATLAB Command 2: c20 = 0, c21 = 0, c22 = 0, c30 = 0, c31 = 0, c32 = 0,
c33 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0, c47 = 0
rank = 39,

∗

0

0

0
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗


0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ ∗ ∗ ∗

0 0 0 0 0 0 ∗ ∗ ∗
MATLAB Command 2: c20 = 0, c21 = 0, c22 = 0, c30 = 0, c31 = 0, c32 = 0,
c33 = 0, c34 = 0, c35 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0,
c46 = 0
26
rank = 39,

∗

0

0

0
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗


0 0 0 ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ ∗ ∗ ∗
MATLAB Command 2: c20 = 0, c21 = 0, c22 = 0, c23 = 0, c30 = 0, c31 = 0,
c32 = 0, c33 = 0, c34 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0
rank = 35,

∗

0

0

0
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗


∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 0 0 0 ∗
MATLAB Command 2: c20 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c34 = 0,
c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0, c47 = 0, c48 = 0
rank = 39,

∗

0

0

0
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗


0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 0 0 0 ∗
MATLAB Command 2: c20 = 0, c21 = 0, c30 = 0, c31 = 0, c32 = 0, c40 = 0,
c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0, c47 = 0, c48 = 0
rank = 39,
27

∗

0

0

0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ 0 ∗ ∗

0 0 0 0 0 0 0 ∗ ∗
MATLAB Command 2: c20 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c34 = 0,
c35 = 0, c37 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0,
c47 = 0
rank = 36,

∗

0

0

0
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗


0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 ∗ 0 0 ∗ ∗ ∗ ∗

0 0 0 0 0 0 0 ∗ ∗
MATLAB Command 2: c20 = 0, c21 = 0, c30 = 0, c31 = 0, c32 = 0, c34 = 0,
c35 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0, c47 = 0
rank = 39,

∗

0

0

0
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗


0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 ∗ 0 0 0 0 ∗ ∗

0 0 0 0 0 ∗ 0 ∗ ∗
MATLAB Command 2: c20 = 0, c21 = 0, c30 = 0, c31 = 0, c32 = 0, c34 = 0,
c35 = 0, c36 = 0, c37 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0,
c47 = 0
rank = 36,
28

∗

0

0

0
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗


0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ 0 ∗ ∗
MATLAB Command 2: c20 = 0, c21 = 0, c22 = 0, c30 = 0, c31 = 0, c32 = 0,
c33 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c47 = 0
rank = 39,

∗

0

0

0
0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗


∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 ∗ 0 0 ∗ ∗ ∗ ∗

0 0 0 0 0 0 0 ∗ ∗
MATLAB Command 2: c11 = 0, c20 = 0, c30 = 0, c31 = 0, c32 = 0, c34 = 0,
c35 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0, c47 = 0
rank = 39,

0

0

0

0

∗ 0 0 ∗ ∗ 0 ∗ ∗ ∗

0 ∗ 0 ∗ ∗ 0 ∗ ∗ ∗

0 0 ∗ 0 0 0 0 ∗ ∗

0 0 0 0 0 ∗ 0 ∗ ∗
MATLAB Command 2: c10 = 0, c12 = 0, c13 = 0, c16 = 0, c20 = 0, c21 = 0,
c23 = 0, c26 = 0, c30 = 0, c31 = 0, c32 = 0, c34 = 0, c35 = 0, c36 = 0, c37 = 0,
c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c47 = 0
rank = 36,
29

0

0

0

0

∗ 0 0 0 ∗ 0 ∗ ∗ ∗

0 ∗ 0 0 0 0 ∗ ∗ ∗

0 0 ∗ 0 0 0 ∗ ∗ ∗

0 0 0 0 0 ∗ 0 ∗ ∗
MATLAB Command 2: c10 = 0, c12 = 0, c13 = 0, c14 = 0, c16 = 0, c20 = 0,
c21 = 0, c23 = 0, c24 = 0, c25 = 0, c26 = 0, c30 = 0, c31 = 0, c32 = 0, c34 = 0,
c35 = 0, c36 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c47 = 0
rank = 38,

0

0

0

0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 0 ∗ ∗ ∗
MATLAB Command 2: c10 = 0, c20 = 0, c21 = 0, c30 = 0, c31 = 0, c32 = 0,
c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0
rank = 39,

0

0

0

0

∗ 0 0 0 ∗ 0 ∗ ∗ ∗

0 0 ∗ 0 0 0 ∗ ∗ ∗

0 0 0 ∗ 0 0 ∗ ∗ ∗

0 0 0 0 0 ∗ ∗ ∗ ∗
MATLAB Command 2: c10 = 0, c12 = 0, c13 = 0, c14 = 0, c16 = 0, c20 = 0,
c21 = 0, c22 = 0, c24 = 0, c25 = 0, c26 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0,
c35 = 0, c36 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0
rank = 38,
30

∗

0

0

0

∗ ∗ 0 0 ∗ 0 ∗ ∗ ∗

0 0 ∗ 0 ∗ 0 ∗ ∗ ∗

0 0 0 ∗ ∗ 0 ∗ ∗ ∗

0 0 0 0 0 ∗ 0 ∗ ∗
MATLAB Command 2: c13 = 0, c14 = 0, c16 = 0, c20 = 0, c21 = 0, c22 = 0,
c24 = 0, c26 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c36 = 0, c40 = 0, c41 = 0,
c42 = 0, c43 = 0, c44 = 0, c45 = 0, c47 = 0
rank = 39,

∗

0

0

0

∗ ∗ ∗ ∗ 0 0 ∗ ∗ ∗

0 0 ∗ ∗ 0 0 ∗ ∗ ∗

0 0 0 0 ∗ 0 ∗ ∗ ∗

0 0 0 0 0 ∗ 0 ∗ ∗
MATLAB Command 2: c15 = 0, c16 = 0, c20 = 0, c21 = 0, c22 = 0, c25 = 0,
c26 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c34 = 0, c36 = 0, c40 = 0, c41 = 0,
c42 = 0, c43 = 0, c44 = 0, c45 = 0, c47 = 0
rank = 39,

∗

0

0

0

0 ∗ 0 ∗ ∗ 0 ∗ ∗ ∗

∗ ∗ 0 ∗ ∗ 0 ∗ ∗ ∗

0 0 ∗ 0 0 0 0 ∗ ∗

0 0 0 0 0 ∗ 0 ∗ ∗
MATLAB Command 2: c11 = 0, c13 = 0, c16 = 0, c20 = 0, c23 = 0, c26 = 0,
c30 = 0, c31 = 0, c32 = 0, c34 = 0, c35 = 0, c36 = 0, c37 = 0, c40 = 0, c41 = 0,
c42 = 0, c43 = 0, c44 = 0, c45 = 0, c47 = 0
rank = 36.
31
The relative invariant of the prehomogeneous space which is relevant here was
computed by Kimura and Sato in [10]. We used its explicit form computed as in [10]
and checked (using the MATLAB code from Appendix E) that each of the loci above
indeed lies in the complement of the general G-orbit (i.e. the rank of the matrix
A associated in Appendix E to a matrix representing the general element of such a
locus is less than 40).
From the output of the Excel macros from Appendix D one sees that the following
Plücker coordinates Zijkl = Zi ∧ Zj ∧ Zk ∧ Zl , i < j < k < l of a 4-dimensional linear
subspace of P(∧2 V ) ∼
= P9 with non-positive weight always vanish (i.e. their weight
is always positive):
Z0123 , Z0124 , Z0125 , Z0126 , Z0127 , Z0128 , Z0129 , Z0134 , Z0135 , Z0137 ,
Z0145 , Z0146 , Z0147 , Z0148 , Z0149 , Z0156 , Z0157 , Z0158 , Z0159 , Z0167 ,
Z0234 , Z0235 , Z0237 , Z0245 , Z0246 , Z0247 , Z0248 , Z0249 , Z0256 , Z0257 , Z0258 ,
Z0267 , Z0345 , Z0347 , Z0357 , Z1234 , Z1235 , Z1245 , Z1246 , Z1256 , Z1345 , Z2345 .
Moreover, if Z1236 does not vanish, then Z1237 = Z1267 = 0. If Z1346 6= 0, then
Z1347 = 0. If Z1356 6= 0, then Z1357 = 0. If Z2346 6= 0, then Z2347 = 0.
Every such 4-dimensional linear subspace lies in one of the loci in G(4,
V2
V)
listed above. Hence it does not lie in the general orbit of G.
This shows that every element of the general G-orbit in G(4,
V2
V ) is properly
stable. Hence the Lemma follows. QED
This proves the following theorem, which is the main result of our thesis.
Theorem 1. Suppose t > 0 is large enough. Then there exists a quasi-projective
quotient space X//Lt G as described in the Problem in the Introduction. Moreover,
semistable and properly stable loci coincide and contain exactly double anticanonical
curves on the smooth del Pezzo surface of degree 5.
Appendix
A
Classification of codimension 4 linear
sections of the Grassmannian of lines
in P4
In this appendix we interpret a result of Ozeki [14] and give an explicit geometric
classification of all codimension 4 linear sections of the Grassmannian of lines in P4 .
The problem of classifying linear sections of the Grassmannian of lines in P4 was
addressed earlier by J.A. Todd [17]. We believe that our results complement his
classification of codimension 4 linear sections.
G = SL(5)-orbits in G(4,
V2
V ) were classified algebraically by Ozeki [14]. They
are listed in Proposition 2.1 in [14]. For each orbit Ozeki provides its explicit representative, so that one has explicit equations, which can be used in order to describe
explicitly the corresponding linear section of the Grassmannian. We list these descriptions in Figures A.1-A.7. The (elementary) method which we use is illustrated
in the case of orbit number 2.
This orbit is the only orbit of codimension 1. Let us check that it can be characterized by the property that the intersection of codimension 4 linear subspaces of
P9 in this orbit with the Grassmannian G(2, 5) ⊂ P9 is a del Pezzo surface with one
ordinary double point. Proposition 2.1 from [14] says that as a representative of this
33
orbit one can take equations
Z9 = 0,
Z6 = −Z1 ,
Z4 = −Z2 ,
Z5 = −Z3 .
If one uses equations p1 = p2 = p3 = p4 = p5 = 0 for the Grassmannain G(2, 5) ⊂ P9 ,
then taking intersection one obtains a surface in P5 with coordinates
Z0 , Z1 , Z2 , Z3 , Z7 , Z8
given by the intersection of quadrics
Z0 Z7 + Z1 Z3 − Z22 , Z0 Z8 − Z2 Z3 + Z12 , Z32 − Z1 Z2 , Z2 Z8 − Z3 Z7 , Z3 Z8 − Z1 Z7 .
After changing coordinates in P5 one obtains equations shown on the figures A.1-A.7
below. It is straightforward to see that this surface S is nondegenerate irreducible
and smooth except for one ordinary double point. Let us find all the lines on S.
The intersection of S with hyperplane Z5 = 0 consists of three lines and a conic
intersecting at the unique point - the singular point on S. All other lines on S are
also lines in its affine chart Z5 = 1. Let us denote by
X0 =
Z0
,
Z5
X1 =
Z1
,
Z5
X2 =
Z2
,
Z5
X3 =
Z3
,
Z5
X4 =
Z4
Z5
the coordinates in this affine chart. Then any line in this chart can be given parametrically as follows:
Xi = ai + bi · t,
i = 0, 1, 2, 3, 4,
where t is a parameter and ai , bi are constants.
Surface S in this affine chart is described by the equations
X3 = X1 X4 , X2 = X1 X42 , X0 = X12 X43 − X12 .
34
They give the following conditions on constants ai , bi :
b1 b4 = 0, a1 b4 = 0, b1 (a34 − 1) = 0.
If b1 6= 0, then we get 3 lines parametrized by η = a4 . If b4 6= 0, then we get one
more line Z0 = Z1 = Z2 = Z3 = 0. From the well-known classification of surfaces of
degree n in Pn it follows that S is a del Pezzo surface of degree 5.
Similarly one checks the other entries of the Figures A.1-A.7.
35
Orbit #
(acc.to
Ozeki
[14])
Codimof
Ozeki’s
orbit
(from
[14])
1
0
picture
description
equations in
with coordinates
A smooth del Pezzo surface of degree
5.
10 lines: (
(
(
)
)
),
where
2
1
A del Pezzo surface of degree 5 with
one singular point of type .
7 lines: (
(
(
where
1
point: (
3
2
2
3
),
)
)
)
)
)
)
)
A del Pezzo surface of degree 5 with
one singular point of type .
4 lines: (
(
(
(
1
point: (
5
)
A del Pezzo surface of degree 5 with
two singular points of type .
4 lines: (
(
(
(
2
points: (
(
4
)
)
)
)
)
)
A del Pezzo surface of degree 5 with
one singular point of type
and one
singular point of type .
3 lines: (
(
(
1
point: (
1
point: (
)
)
)
Figure A.1. Codimension 4 sections of G(2, 5) ⊂ P9 , orbits 1-5.
)
)
36
Orbit #
(acc.to
Ozeki
[14])
Codimof
Ozeki’s
orbit
(from
[14])
6
3
7
4
picture
description
equations in
with coordinates
A del Pezzo surface of degree 5 with
one singular point of type .
2 lines: (
(
1
point: (
A nonsingular quadric surface and a Quadric surface:
nonsingular cubic scroll intersecting (
along a conic which is a directrix of the Scroll:
scroll.
)
)
)
=
)
Conic of intersection:
(
=
)
The second directrix of the scroll:
(
)
8
4
9
4
10
5
11
5
A projection of a quintic scroll in
(whose directrices are a conic and a
twisted cubic) from a point lying in the
plane generated by the conic. The
projected scroll is singular along a line
)
(the image of the conic) which is a 1 double line: (
The second directrix (a twisted cubic):
directrix.
(
=
)
A del Pezzo surface of degree 5 with
one singular point of type .
1 line: (
)
1
point: (
)
A quartic scroll in
and a plane Plane:
( =
)
intersecting along a conic which is a
Conic of intersection:
directrix of the scroll.
(
=
)
The second directrix of the scroll (also a conic):
(
)
A cubic scroll and a quadric cone Quadric cone:
intersecting along a nonsingular conic
(
=
)
which is their common directrix.
Scroll:
Conic of intersection:
(
=
)
The second directrix of the scroll:
(
)
Figure A.2. Codimension 4 sections of G(2, 5) ⊂ P9 , orbits 6-11.
37
Orbit #
(acc.to
Ozeki
[14])
Codimof
Ozeki’s
orbit
(from
[14])
12
5
13
5
14
6
15
6
16
6
17
6
picture
description
equations in
with coordinates
A nonsingular quadric surface and a Quadric surface:
nonsingular cubic scroll intersecting (
=
)
along a pair of lines one of which is a Scroll:
directrix of the scroll and the other is a
ruling. The second directrix of the
scroll is a conic intersecting the
quadric at one point.
2 lines of intersection:
(
) (directrix)
(
) (ruling)
The second directrix of the scroll (a conic):
(
=
)
A projection of a quintic scroll in
(whose directrices are a line and a
rational normal quartic) from a point
lying in the plane generated by two
lines – a directrix and a ruling. The
)
projected scroll is singular along a line 1 double line: (
(the image of the directrix) which is a The second directrix (a rational normal quartic):
(
directrix.
), where t is a parameter.
A plane and a Veronese surface
Plane:
( =
)
intersecting along a conic.
Conic of intersection:
(
=
)
A pair of nonsingular quadric surfaces
and a plane. Each pair of surfaces
intersects along a line. The three lines
of intersection intersect at the unique
point.
Plane:
(
2 quadrics:
(
(
=
)
=
=
)
)
A plane and a nonsingular quartic Plane:
scroll intersecting along a pair of lines
( =
)
– a ruling and a directrix of the scroll.
2 lines of intersection:
(
) (directrix)
(
) (ruling)
The second directrix (a twisted cubic):
(
=
)
A nonsingular quadric surface and a Quadric surface: (
=
cone over a twisted cubic intersecting Scroll:
along a pair of lines – two rulings.
2 lines of intersection:
(
(
Vertex: ( =
Twisted cubic: (
)
)
)
)
Figure A.3. Codimension 4 sections of G(2, 5) ⊂ P9 , orbits 12-17.
)
=
38
Orbit #
(acc.to
Ozeki
[14])
Codimof
Ozeki’s
orbit
(from
[14])
18
6
19
7
20
7
21
7
22
8
picture
description
equations in
A two-dimensional quadric cone and a Quadric cone:
cubic scroll intersecting along two lines (
=
which are rulings of the cone – a ruling Scroll:
and a directrix of the scoll.
A pair of planes and a cubic scroll. Each
pair of surfaces intersects along a line.
The scroll intersects with one plane
along a ruling and with the other plane
along a directrix. The three lines
intersect at one point.
with coordinates
)
2 lines of intersection:
(
) (directrix of the scroll)
(
) (ruling of the scroll)
The second directrix of the scroll (a conic):
(
=
)
2 planes:
( =
)
( =
)
Scroll:
2 lines of intersection of the scroll with planes:
(
) (directrix)
(
) (ruling)
The second directrix of the scroll (a conic):
(
=
)
A nonsingular quadric surface, a 2 quadrics:
quadric cone and a plane. Each pair of (
=
)
surfaces intersects along a line. Three (
=
)
lines intersect at one point which is
the vertex of the cone.
Plane: (
)
A quadric cone and a cone over a
twisted cubic intersecting along a
common ruling. The line of
intersection passes through vertices of
both cones.
A plane (with multiplicity 1) and a
nonsingular quadric surface (with
multiplicity 2) intersecting along a line.
Quadric cone:
(
=
)
Twisted cubic:
(
=
)
Vertex of the cubic cone:
(
)
1 line of intersection:
(
)
Quadric:
(
=
)
Plane: (
)
Figure A.4. Codimension 4 sections of G(2, 5) ⊂ P9 , orbits 18-22.
=
39
Orbit #
(acc.to
Ozeki
[14])
Codimof
Ozeki’s
orbit
(from
[14])
23
8
24
25
26
27
28
8
9
9
9
10
picture
description
Three planes and a nonsingular
quadric surface. Two of the planes
intersect at the unique point, which is
the point of intersection of the quadric
with the third plane. The third plane
and the quadric intersect each of the
other two planes along a line. The four
lines of intersection intersect at the
point above.
Two quadric cones and a plane. The
cones touch each other along a
common ruling. The plane touches
both cones along this ruling.
Four planes one of which has
multiplicity 2. The plane with
multiplicity 2 intersects each of the
other planes along a line. Each pair of
planes with multiplicity 1 intersects at
a point. The three points of
intersection are the three points of the
lines of intersection and determine the
multiplicity 2 plane.
A two-dimensional quadric cone and
two planes one of which has
multiplicity 2. The cone intersects the
multiplicity 2 plane along a line and
the other plane at a point. The planes
intersect along a line. The lines
intersect at the point above.
equations in
Quadric:
(
)
)
)
)
=
=
)
)
Plane: (
)
4 planes:
(
(
(
(
)
)
)
) (multiplicity 2)
2 planes:
(
(
) (multiplicity 2)
)
Cone:
(
A quadric cone (with multiplicity 2) Plane:
and a plane which touches the cone (
along its ruling.
Cone:
(
Three plane (two of which have
multiplicity 2). One of the planes with
multiplicity 2 intersects each of the
other two planes along a line. The lines
intersect at a point which is the point
of intersection of the other two
planes.
=
3 planes: (
(
(
2 cones:
(
(
with coordinates
3 planes:
(
(
(
=
)
)
=
)
) (multiplicity 2)
) (multiplicity 2)
)
Figure A.5. Codimension 4 sections of G(2, 5) ⊂ P9 , orbits 23-28.
40
Orbit #
(acc.to
Ozeki
[14])
Codimof
Ozeki’s
orbit
(from
[14])
29
11
30
11
picture
description
equations in
with coordinates
A
nonsingular
three-dimensional Plane:
quadric and a plane intersecting along (
a line.
Quadric:
(
=
Two planes – one with multiplicity 2
and another with multiplicity 3. The
planes intersect along a line.
2 planes:
(
(
) (multiplicity 2)
) (multiplicity 3)
)
)
31
12
Two disjoint planes and a threedimensional linear subspace, which
intersects each of the planes along a
line.
2 planes:
(
)
(
)
Three-dimensional linear subspace:
(
)
32
12
A quadric cone of dimension 3 with
zero-dimensional vertex and a plane
intersecting along a line.
Plane:
(
)
Cone:
(
33
12
34
13
=
Segre embedding
- a nondegenerate smooth irreducible cubic
threefold.
A three-dimensional linear subspace
Plane:
and a plane intersecting along a line.
(
)
)
Three-dimensional linear subspace:
(
)
35
13
A three-dimensional cone over a cubic
scroll.
Vertex:
(
)
Scroll:
Directrices of the scroll (a line and a conic):
(
)
(
)
Figure A.6. Codimension 4 sections of G(2, 5) ⊂ P9 , orbits 29-35.
41
Orbit #
(acc.to
Ozeki
[14])
Codimof
Ozeki’s
orbit
(from
[14])
36
14
37
15
38
20
picture
description
A quadric cone of dimension 3 with
the zero-dimensional vertex. The
linear section itself is not reduced.
A quadric cone of dimension 3 with
the zero-dimensional vertex and a
three-dimensional linear subspace
intersecting along a plane.
equations in
with coordinates
Three-dimensional linear subspace:
=0)
Cone:
(
)
A nonsingular quadric of dimension 4.
Figure A.7. Codimension 4 sections of G(2, 5) ⊂ P9 , orbits 36-38.
Appendix
B
Grassmannian degenerations of the
del Pezzo quintic threefold
In this appendix we list all codimension 3 linear sections of the Grassmannian of
lines in P4 . The problem of classifying linear sections of the Grassmannian of lines
in P4 was addressed earlier by J.A. Todd [17]. In particular, in [17] he obtained the
complete classification of linear sections of codimension 3. It is now well-known that
a general such section is the del Pezzo quintic threefold. Computations are analogous
to those which we used in Appendix A. Instead of the orbit classification of Ozeki [14],
we use the classification of orbits of another prehomogeneous space due to Kimura [9].
The define a del Pezzo quintic threefold to be a nondegenerate irreducible threefold
of degree 5 in P6 with only finitely many planes which is smoothable to a nonsingular
del Pezzo quintic threefold. We call a threefold singularity a pinch point, if it is locally
isomorphic to
x2 + y 2 + zw2 = 0,
and a higher pinch point, if it is locally isomorphic to
x2 + y 2 + zw3 = 0.
43
Orbit #
(acc.to
Kimura
[9])
Codim of
Kimura’s
orbit
(from
[9])
1
0
A smooth del Pezzo quintic threefold.
2
1
A del Pezzo quintic threefold with one
singular point of type .
picture
equations in
description
with coordinates
No planes
1
point: (
)
No planes
3
2
A del Pezzo quintic threefold with two
singular points of type .
2
(
(
points:
)
)
1 plane: (
4
3
A del Pezzo quintic threefold with
three singular points of type
.The
threefold contains four planes. One
plane is determined by the singular
points and each singular point is its
intersection with a pair of two other 3
(
planes.
(
(
points:
4 planes:
(
(
(
(
5
3
)
)
)
)
)
)
)
)
A del Pezzo quintic threefold with one
double
line,
which
intersects
transversally the unique plane on the
threefold at a pinch point.
The double line:
(
)
1 plane: (
Figure B.1. Codimension 3 sections of G(2, 5) ⊂ P9 , orbits 1-5.
)
44
Orbit #
(acc.to
Kimura
[9])
Codim of
Kimura’s
orbit
(from
[9])
6
4
7
8
5
6
picture
equations in
description
A del Pezzo quintic threefold with one
double line and one isolated singular
point of type
.The threefold
contains three planes. One plane is
determined by the double line and the
double point. Another plane intersects
it along a line passing through the
double point and the point of
intersection of all three planes with
the double line, which is a pinch point
on the threefold.
with coordinates
The double line:
(
1
)
point: (
3 planes: (
(
(
)
)
)
)
A del Pezzo quintic threefold with one
triple line. The threefold contains two
planes. One plane contains the triple
line and intersects the other plane
along a transversal line. The lines
The triple line:
intersect at a higher pinch point.
(
2 planes: (
(
A nonsingular quadric threefold and a Quadric threefold:
Segre
threefold
intersecting along a smooth quadric Segre threefold scroll:
surface, which is a directrix of the
Segre plane scroll.
)
)
)
Quadric surface:
The second directrix of the scroll (a line):
9
7
A three-dimensional quadric cone and Quadric cone:
Segre
threefold
intersecting along a smooth quadric Segre threefold scroll:
surface, which is a directrix of the
Segre plane scroll and the base of the
cone.
Quadric surface:
The second directrix of the scroll (a line):
10
7
A projection of a plane quintic scroll in
, whose directrices are a smooth
quadric surface (which projects onto
the double plane) and a twisted cubic
from a point in the space determined
The double plane:
by the quadric.
(
The twisted cubic:
(
)
)
Figure B.2. Codimension 3 sections of G(2, 5) ⊂ P9 , orbits 6-10.
45
Orbit #
(acc.to
Kimura
[9])
Codim of
Kimura’s
orbit
(from
[9])
11
7
picture
description
equations in
with coordinates
A nonsingular quadric threefold and a Quadric threefold:
cone over a nonsingular cubic surface
scroll intersecting along a twodimensional quadric cone which Cubic scroll:
shares a directix with the scroll and the
vertex with the cubic cone.
Vertex: (
)
Quadric cone:
Directrices of the scroll:
(a line)
(a conic)
12
8
A three-dimensional linear space and a Linear space:
blowup of
at a line embedded into
by the complete linear system |2HE| (where E=
is the Quartic scroll:
exceptional divisor and H is a total
transform of a plane in
) – a
nonsingular plane threefold scroll of
degree 4. The space and the scroll
intersect along a smooth quadric
surface which is the image of the
exceptional divisor and is a directrix of Quadric surface:
the scroll.
The second directrix of the scroll (a conic):
14
8
A three-dimensional quadric cone and Quadric cone:
a cone over a nonsingular cubic
surface scroll intersecting along two
planes – rulings of the quadric cone. Scroll:
One of the planes intersects the linear
subspace generated by the scroll along
a directrix of the scroll, the other plane
– along a ruling of the scroll.
Directrices of the scroll:
(a line)
(a conic)
Vertex of the cubic cone:
(
2 planes:
Figure B.3. Codimension 3 sections of G(2, 5) ⊂ P9 , orbits 11-14.
)
46
Orbit #
(acc.to
Kimura
[9])
Codim of
Kimura’s
orbit
(from
[9])
15
9
17
11
picture
description
A three-dimensional linear space and a
pair of three-dimensional quadric
cones with zero-dimensional vertices.
Each pair of the threefolds intersects
along a plane in a pencil of planes
determined by the line passing
through the vertices of the cones.
A three-dimensional linear space (with
multiplicity 1) and a non-reduced
three-dimensional quadric cone (with
multiplicity 2) intersecting along (a
non-reduced) plane.
equations in
with coordinates
2 quadric cones:
Linear space:
:
Linear space:
:
Quadric cone (after reduction):
Plane:
20
21
14
15
A nonsingular quadric fourfold and a Quadric:
three-dimensional
linear
space
intersecting along a plane.
Linear space:
A cone over a Segre threefold Vertex:
(
.
:
)
The linear subspace generated by the Segre
threefold:
:
Figure B.4. Codimension 3 sections of G(2, 5) ⊂ P9 , orbits 15-21.
Appendix
C
Excel macros which computes
extremal destabilizing 1-PS
In this appendix we provide the Excel macros, which we used in Chapter 3 in order
to compute extremal destabilizing 1-PS. We used Microsoft Visual Basic 6.5 included
in Microsoft Office Excel 2007.
Sub MacrosComputingDestabilizing1PS()
’List variables.
Dim i, j As Integer
Dim a1, a2, a3 As Integer
Dim c(1 To 4) As Integer
Dim Eqns(1 To 100, 1 To 4) As Integer ’The second index is the coefficient, the first
index is the number of the equation.
Dim Triples(1 To 3, 1 To 4) As Integer
Dim Solutions(1 To 10000, 1 To 4) As Integer ’Stores solutions.
Dim count As Integer ’Counts solutions.
Dim Det As Integer
Dim DetA(1 To 3) As Integer
Dim ExtraSolutions(1 To 10000, 1 To 4) As Integer ’Stores solutions with c4=0.
’Data Input.
48
’We list 4-tuples (u1,u2,u3,u4) except for the 4-tuple (0,0,0,1) (corresponding to the
equation c4=0). Solutions with c4=0 were computed separately - see below.
Eqns(1, 1) = 1: Eqns(1, 2) = -1: Eqns(1, 3) = 0: Eqns(1, 4) = 0:
Eqns(2, 1) = 1: Eqns(2, 2) = 0: Eqns(2, 3) = -1: Eqns(2, 4) = 0:
Eqns(3, 1) = 1: Eqns(3, 2) = 0: Eqns(3, 3) = 0: Eqns(3, 4) = -1:
Eqns(4, 1) = 0: Eqns(4, 2) = 1: Eqns(4, 3) = -1: Eqns(4, 4) = 0:
Eqns(5, 1) = 0: Eqns(5, 2) = 1: Eqns(5, 3) = 0: Eqns(5, 4) = -1:
Eqns(6, 1) = 0: Eqns(6, 2) = 0: Eqns(6, 3) = 1: Eqns(6, 4) = -1:
Eqns(7, 1) = 2: Eqns(7, 2) = 0: Eqns(7, 3) = 0: Eqns(7, 4) = -1:
Eqns(8, 1) = 1: Eqns(8, 2) = 0: Eqns(8, 3) = 0: Eqns(8, 4) = -2:
Eqns(9, 1) = 2: Eqns(9, 2) = 0: Eqns(9, 3) = -1: Eqns(9, 4) = 0:
Eqns(10, 1) = 1: Eqns(10, 2) = 0: Eqns(10, 3) = -2: Eqns(10, 4) = 0:
Eqns(11, 1) = 0: Eqns(11, 2) = 2: Eqns(11, 3) = 0: Eqns(11, 4) = -1:
Eqns(12, 1) = 0: Eqns(12, 2) = 1: Eqns(12, 3) = 0: Eqns(12, 4) = -2:
Eqns(13, 1) = 0: Eqns(13, 2) = 1: Eqns(13, 3) = 1: Eqns(13, 4) = -1:
Eqns(14, 1) = 0: Eqns(14, 2) = 1: Eqns(14, 3) = -1: Eqns(14, 4) = -1:
Eqns(15, 1) = 1: Eqns(15, 2) = -1: Eqns(15, 3) = -2: Eqns(15, 4) = 0:
Eqns(16, 1) = 2: Eqns(16, 2) = 1: Eqns(16, 3) = -1: Eqns(16, 4) = 0:
Eqns(17, 1) = 1: Eqns(17, 2) = -1: Eqns(17, 3) = 1: Eqns(17, 4) = 0:
Eqns(18, 1) = 1: Eqns(18, 2) = 0: Eqns(18, 3) = -2: Eqns(18, 4) = -2:
Eqns(19, 1) = 2: Eqns(19, 2) = 0: Eqns(19, 3) = -1: Eqns(19, 4) = -1:
Eqns(20, 1) = 1: Eqns(20, 2) = 2: Eqns(20, 3) = 0: Eqns(20, 4) = -2:
Eqns(21, 1) = 1: Eqns(21, 2) = -1: Eqns(21, 3) = 0: Eqns(21, 4) = 2:
Eqns(22, 1) = 1: Eqns(22, 2) = -1: Eqns(22, 3) = 0: Eqns(22, 4) = -2:
Eqns(23, 1) = 2: Eqns(23, 2) = 1: Eqns(23, 3) = 0: Eqns(23, 4) = -2:
Eqns(24, 1) = 1: Eqns(24, 2) = 0: Eqns(24, 3) = -1: Eqns(24, 4) = -2:
Eqns(25, 1) = 1: Eqns(25, 2) = 0: Eqns(25, 3) = 1: Eqns(25, 4) = -1:
Eqns(26, 1) = 1: Eqns(26, 2) = 0: Eqns(26, 3) = -1: Eqns(26, 4) = 1:
Eqns(27, 1) = 1: Eqns(27, 2) = 2: Eqns(27, 3) = 0: Eqns(27, 4) = -1:
Eqns(28, 1) = 1: Eqns(28, 2) = 1: Eqns(28, 3) = 0: Eqns(28, 4) = -1:
Eqns(29, 1) = 1: Eqns(29, 2) = 0: Eqns(29, 3) = -1: Eqns(29, 4) = -1:
Eqns(30, 1) = 1: Eqns(30, 2) = 0: Eqns(30, 3) = -2: Eqns(30, 4) = -1:
Eqns(31, 1) = 1: Eqns(31, 2) = -1: Eqns(31, 3) = 0: Eqns(31, 4) = 1:
49
Eqns(32, 1) = 1: Eqns(32, 2) = -1: Eqns(32, 3) = 0: Eqns(32, 4) = -1:
Eqns(33, 1) = 2: Eqns(33, 2) = 1: Eqns(33, 3) = 0: Eqns(33, 4) = -1:
Eqns(34, 1) = 2: Eqns(34, 2) = 0: Eqns(34, 3) = -1: Eqns(34, 4) = -2:
Eqns(35, 1) = 2: Eqns(35, 2) = 0: Eqns(35, 3) = 1: Eqns(35, 4) = -1:
Eqns(36, 1) = 2: Eqns(36, 2) = 0: Eqns(36, 3) = -1: Eqns(36, 4) = 1:
Eqns(37, 1) = 2: Eqns(37, 2) = 0: Eqns(37, 3) = -2: Eqns(37, 4) = -1:
Eqns(38, 1) = 1: Eqns(38, 2) = 1: Eqns(38, 3) = 0: Eqns(38, 4) = -2:
Eqns(39, 1) = 2: Eqns(39, 2) = 2: Eqns(39, 3) = 0: Eqns(39, 4) = -1:
Eqns(40, 1) = 0: Eqns(40, 2) = 1: Eqns(40, 3) = -1: Eqns(40, 4) = 1:
Eqns(41, 1) = 0: Eqns(41, 2) = 1: Eqns(41, 3) = -1: Eqns(41, 4) = -2:
Eqns(42, 1) = 0: Eqns(42, 2) = 2: Eqns(42, 3) = 1: Eqns(42, 4) = -1:
Eqns(43, 1) = 1: Eqns(43, 2) = 1: Eqns(43, 3) = -1: Eqns(43, 4) = 0:
Eqns(44, 1) = 1: Eqns(44, 2) = -1: Eqns(44, 3) = -1: Eqns(44, 4) = 0:
Eqns(45, 1) = 1: Eqns(45, 2) = -1: Eqns(45, 3) = 1: Eqns(45, 4) = 2:
Eqns(46, 1) = 2: Eqns(46, 2) = 1: Eqns(46, 3) = -1: Eqns(46, 4) = 1:
Eqns(47, 1) = 1: Eqns(47, 2) = -1: Eqns(47, 3) = 1: Eqns(47, 4) = -1:
Eqns(48, 1) = 2: Eqns(48, 2) = 1: Eqns(48, 3) = -1: Eqns(48, 4) = -2:
Eqns(49, 1) = 1: Eqns(49, 2) = 3: Eqns(49, 3) = 1: Eqns(49, 4) = -1:
Eqns(50, 1) = 1: Eqns(50, 2) = -1: Eqns(50, 3) = -3: Eqns(50, 4) = -1:
Eqns(51, 1) = 2: Eqns(51, 2) = 3: Eqns(51, 3) = 1: Eqns(51, 4) = -1:
Eqns(52, 1) = 1: Eqns(52, 2) = -1: Eqns(52, 3) = -3: Eqns(52, 4) = -2:
Eqns(53, 1) = 1: Eqns(53, 2) = 2: Eqns(53, 3) = 1: Eqns(53, 4) = -1:
Eqns(54, 1) = 1: Eqns(54, 2) = -1: Eqns(54, 3) = -2: Eqns(54, 4) = -1:
Eqns(55, 1) = 1: Eqns(55, 2) = 1: Eqns(55, 3) = 1: Eqns(55, 4) = -1:
Eqns(56, 1) = 1: Eqns(56, 2) = -1: Eqns(56, 3) = -1: Eqns(56, 4) = -1:
Eqns(57, 1) = 1: Eqns(57, 2) = 1: Eqns(57, 3) = -1: Eqns(57, 4) = -1:
Eqns(58, 1) = 1: Eqns(58, 2) = -1: Eqns(58, 3) = -2: Eqns(58, 4) = -2:
Eqns(59, 1) = 2: Eqns(59, 2) = 2: Eqns(59, 3) = 1: Eqns(59, 4) = -1:
Eqns(60, 1) = 2: Eqns(60, 2) = 1: Eqns(60, 3) = -1: Eqns(60, 4) = -1:
Eqns(61, 1) = 1: Eqns(61, 2) = 1: Eqns(61, 3) = -1: Eqns(61, 4) = -2:
Eqns(62, 1) = 1: Eqns(62, 2) = -1: Eqns(62, 3) = 1: Eqns(62, 4) = 1:
Eqns(63, 1) = 1: Eqns(63, 2) = 1: Eqns(63, 3) = -1: Eqns(63, 4) = 1:
Eqns(64, 1) = 1: Eqns(64, 2) = -1: Eqns(64, 3) = -1: Eqns(64, 4) = -2:
50
Eqns(65, 1) = 2: Eqns(65, 2) = 1: Eqns(65, 3) = 1: Eqns(65, 4) = -1:
Eqns(66, 1) = 1: Eqns(66, 2) = -1: Eqns(66, 3) = -1: Eqns(66, 4) = 1:
Eqns(67, 1) = 1: Eqns(67, 2) = 0: Eqns(67, 3) = 0: Eqns(67, 4) = 0:
Eqns(68, 1) = 0: Eqns(68, 2) = 1: Eqns(68, 3) = 0: Eqns(68, 4) = 0:
Eqns(69, 1) = 0: Eqns(69, 2) = 0: Eqns(69, 3) = 1: Eqns(69, 4) = 0:
’Computation.
count = 0
For a1 = 1 To 67
For a2 = a1 + 1 To 68
For a3 = a2 + 1 To 69
’Read off a new triple of equations.
For j = 1 To 4
Triples(1, j) = Eqns(a1, j)
Triples(2, j) = Eqns(a2, j)
Triples(3, j) = Eqns(a3, j)
Next j
Det = Triples(1, 1) * Triples(2, 2) * Triples(3, 3) - Triples(1, 1) * Triples(2, 3) *
Triples(3, 2) + Triples(1, 2) * Triples(2, 3) * Triples(3, 1) - Triples(1, 2) * Triples(2,
1) * Triples(3, 3) + Triples(1, 3) * Triples(2, 1) * Triples(3, 2) - Triples(1, 3) *
Triples(2, 2) * Triples(3, 1)
If Det hi 0 Then
’Compute the solution.
DetA(1) = Triples(1, 4) * Triples(2, 2) * Triples(3, 3) - Triples(1, 4) * Triples(2, 3) *
Triples(3, 2) + Triples(1, 2) * Triples(2, 3) * Triples(3, 4) - Triples(1, 2) * Triples(2,
4) * Triples(3, 3) + Triples(1, 3) * Triples(2, 4) * Triples(3, 2) - Triples(1, 3) *
Triples(2, 2) * Triples(3, 4)
DetA(2) = Triples(1, 1) * Triples(2, 4) * Triples(3, 3) - Triples(1, 1) * Triples(2, 3) *
Triples(3, 4) + Triples(1, 4) * Triples(2, 3) * Triples(3, 1) - Triples(1, 4) * Triples(2,
51
1) * Triples(3, 3) + Triples(1, 3) * Triples(2, 1) * Triples(3, 4) - Triples(1, 3) *
Triples(2, 4) * Triples(3, 1)
DetA(3) = Triples(1, 1) * Triples(2, 2) * Triples(3, 4) - Triples(1, 1) * Triples(2, 4) *
Triples(3, 2) + Triples(1, 2) * Triples(2, 4) * Triples(3, 1) - Triples(1, 2) * Triples(2,
1) * Triples(3, 4) + Triples(1, 4) * Triples(2, 1) * Triples(3, 2) - Triples(1, 4) *
Triples(2, 2) * Triples(3, 1)
’Check for positivity.
c(1) = -DetA(1)
If c(1) * Det h 0 Then GoTo 1
c(2) = -DetA(2)
If c(2) * Det h 0 Then GoTo 1
c(3) = -DetA(3)
If c(3) * Det h 0 Then GoTo 1
c(4) = Det
’Make the solution positive.
If Det h 0 Then
For j = 1 To 4
c(j) = -c(j)
Next j
End If
’Primitivize the solution.
If c(4) i 1 Then
For i = c(4) To 2 Step -1
If c(4) Mod i = 0 Then
If c(3) Mod i = 0 Then
If c(2) Mod i = 0 Then
If c(1) Mod i = 0 Then
For j = 1 To 4
c(j) = Int(c(j) / i)
Next j
52
Exit For
End If
End If
End If
End If
Next i
End If
’Check if this solutions was already found.
If count i 0 Then
For i = 1 To count
If (Solutions(i, 1) = c(1)) And (Solutions(i, 2) = c(2)) And (Solutions(i, 3) = c(3))
And (Solutions(i, 4) = c(4)) Then GoTo 1
Next i
End If
’Add a new solution to the list.
count = count + 1
For j = 1 To 4
Solutions(count, j) = c(j)
Next j
If count i 9000 Then MsgBox (”Alarm!!! Too many solutions!!!”)
1: End If ’ Det hi 0
Next a3
Next a2
Next a1
’Add solutions with c4=0 (these were computed manually by the same method
as above).
ExtraSolutions(count + 1, 1) = 1: ExtraSolutions(count + 1, 2) = 0: ExtraSolutions(count + 1, 3) = 0: ExtraSolutions(count + 1, 4) = 0:
53
ExtraSolutions(count + 2, 1) = 0: ExtraSolutions(count + 2, 2) = 1: ExtraSolutions(count + 2, 3) = 0: ExtraSolutions(count + 2, 4) = 0:
ExtraSolutions(count + 3, 1) = 0: ExtraSolutions(count + 3, 2) = 0: ExtraSolutions(count + 3, 3) = 1: ExtraSolutions(count + 3, 4) = 0:
ExtraSolutions(count + 4, 1) = 0: ExtraSolutions(count + 4, 2) = 1: ExtraSolutions(count + 4, 3) = 1: ExtraSolutions(count + 4, 4) = 0:
ExtraSolutions(count + 5, 1) = 2: ExtraSolutions(count + 5, 2) = 0: ExtraSolutions(count + 5, 3) = 1: ExtraSolutions(count + 5, 4) = 0:
ExtraSolutions(count + 6, 1) = 3: ExtraSolutions(count + 6, 2) = 0: ExtraSolutions(count + 6, 3) = 1: ExtraSolutions(count + 6, 4) = 0:
ExtraSolutions(count + 7, 1) = 1: ExtraSolutions(count + 7, 2) = 0: ExtraSolutions(count + 7, 3) = 1: ExtraSolutions(count + 7, 4) = 0:
ExtraSolutions(count + 8, 1) = 1: ExtraSolutions(count + 8, 2) = 0: ExtraSolutions(count + 8, 3) = 2: ExtraSolutions(count + 8, 4) = 0:
ExtraSolutions(count + 9, 1) = 1: ExtraSolutions(count + 9, 2) = 1: ExtraSolutions(count + 9, 3) = 0: ExtraSolutions(count + 9, 4) = 0:
ExtraSolutions(count + 10, 1) = 1: ExtraSolutions(count + 10, 2) = 1: ExtraSolutions(count + 10, 3) = 1: ExtraSolutions(count + 10, 4) = 0:
ExtraSolutions(count + 11, 1) = 2: ExtraSolutions(count + 11, 2) = 2: ExtraSolutions(count + 11, 3) = 1: ExtraSolutions(count + 11, 4) = 0:
ExtraSolutions(count + 12, 1) = 1: ExtraSolutions(count + 12, 2) = 1: ExtraSolutions(count + 12, 3) = 2: ExtraSolutions(count + 12, 4) = 0:
ExtraSolutions(count + 13, 1) = 1: ExtraSolutions(count + 13, 2) = 1: ExtraSolutions(count + 13, 3) = 3: ExtraSolutions(count + 13, 4) = 0:
ExtraSolutions(count + 14, 1) = 1: ExtraSolutions(count + 14, 2) = 2: ExtraSolutions(count + 14, 3) = 2: ExtraSolutions(count + 14, 4) = 0:
ExtraSolutions(count + 15, 1) = 4: ExtraSolutions(count + 15, 2) = 1: ExtraSolutions(count + 15, 3) = 1: ExtraSolutions(count + 15, 4) = 0:
ExtraSolutions(count + 16, 1) = 2: ExtraSolutions(count + 16, 2) = 3: ExtraSolutions(count + 16, 3) = 1: ExtraSolutions(count + 16, 4) = 0:
ExtraSolutions(count + 17, 1) = 1: ExtraSolutions(count + 17, 2) = 3: ExtraSolutions(count + 17, 3) = 2: ExtraSolutions(count + 17, 4) = 0:
ExtraSolutions(count + 18, 1) = 1: ExtraSolutions(count + 18, 2) = 2: ExtraSolu-
54
tions(count + 18, 3) = 1: ExtraSolutions(count + 18, 4) = 0:
ExtraSolutions(count + 19, 1) = 3: ExtraSolutions(count + 19, 2) = 1: ExtraSolutions(count + 19, 3) = 1: ExtraSolutions(count + 19, 4) = 0:
ExtraSolutions(count + 20, 1) = 2: ExtraSolutions(count + 20, 2) = 1: ExtraSolutions(count + 20, 3) = 1: ExtraSolutions(count + 20, 4) = 0:
’Add extra solutions to the set of all solutions.
For i = 1 To 20
count = count + 1
For j = 1 To 4
Solutions(count, j) = ExtraSolutions(count, j)
Next j
Next i
’Output.
Cells(1, 1).Value = ”c1”
Cells(1, 2).Value = ”c2”
Cells(1, 3).Value = ”c3”
Cells(1, 4).Value = ”c4”
For i = 1 To count
For j = 1 To 4
Cells(i + 1, j).Value = Solutions(i, j)
Next j
Next i
End Sub
Appendix
D
Excel macros which identifies minimal
sets of vanishing Plücker coordinates
for unstable points
In this appendix we provide the Excel macros, which we used in Chapter 4 in order to identify the minimal sets of Plücker coordinates, which have to vanish for an
unstable or properly semistable point. We used Microsoft Visual Basic 6.5 included
in Microsoft Office Excel 2007. We assume that the Excel macros from Appendix C
was already run and its output is written on the Excel worksheet.
Sub MacrosIdentifyingVanishingPluckerCoordinates()
’List variables.
Dim i, j, k As Integer
Dim p1, p2, p3, p4 As Integer
Dim c(1 To 4) As Integer
Dim Solutions(1 To 10000, 1 To 4) As Integer ’Solutions found by the Excel macros
from Appendix C.
Dim Taus(1 To 100000, 0 To 9) As Integer ’The weights of Z0,Z1,Z2,Z3,Z4,Z5,Z6,Z7,
Z8,Z9.
Dim WeightsOfPlane(1 To 10000, 0 To 9, 0 To 9, 0 To 9, 0 To 9) As Integer ’The
weights of the 4-plane.
56
Dim count As Integer
count = 983 ’The total number of solutions obtained by the Excel macros from
Appendix C.
’Data Input.
For i = 1 To count
’Read off the solutions.
For j = 1 To 4
Solutions(i, j) = Cells(i + 1, j).Value
c(j) = Solutions(i, j)
Next j
’Compute the weights of Z0,Z1,Z2,Z3,Z4,Z5,Z6,Z7,Z8,Z9.
Taus(i, 0) = 3 * c(1) + 6 * c(2) + 4 * c(3) + 2 * c(4)
Taus(i, 1) = 3 * c(1) + c(2) + 4 * c(3) + 2 * c(4)
Taus(i, 2) = 3 * c(1) + c(2) - c(3) + 2 * c(4)
Taus(i, 3) = 3 * c(1) + c(2) - c(3) - 3 * c(4)
Taus(i, 4) = (-2) * c(1) + c(2) + 4 * c(3) + 2 * c(4)
Taus(i, 5) = (-2) * c(1) + c(2) - c(3) + 2 * c(4)
Taus(i, 6) = (-2) * c(1) + c(2) - c(3) - 3 * c(4)
Taus(i, 7) = (-2) * c(1) - 4 * c(2) - c(3) + 2 * c(4)
Taus(i, 8) = (-2) * c(1) - 4 * c(2) - c(3) - 3 * c(4)
Taus(i, 9) = (-2) * c(1) - 4 * c(2) - 6 * c(3) - 3 * c(4)
’Compute the weights of the 4-plane.
For p1 = 0 To 6
For p2 = p1 + 1 To 7
For p3 = p2 + 1 To 8
For p4 = p3 + 1 To 9
WeightsOfPlane(i, p1, p2, p3, p4) = Taus(i, p1) + Taus(i, p2) + Taus(i, p3) +
Taus(i, p4)
57
Next p4
Next p3
Next p2
Next p1
Next i
’Prepare the header.
j=0
For p1 = 0 To 6
For p2 = p1 + 1 To 7
For p3 = p2 + 1 To 8
For p4 = p3 + 1 To 9
j=j+1
Cells(1, 5 + j).Value = ”Z” + Str(p1) + Str(p2) + Str(p3) + Str(p4)
Next p4
Next p3
Next p2
Next p1
’List weights of Plücker coordinates for all solutions.
For i = 1 To count
j=0
For p1 = 0 To 6
For p2 = p1 + 1 To 7
For p3 = p2 + 1 To 8
For p4 = p3 + 1 To 9
j=j+1
If WeightsOfPlane(i, p1, p2, p3, p4) i 0 Then Cells(1 + i, 5 + j).Value = 1
If WeightsOfPlane(i, p1, p2, p3, p4) h 0 Then Cells(1 + i, 5 + j).Value = -1
If WeightsOfPlane(i, p1, p2, p3, p4) = 0 Then Cells(1 + i, 5 + j).Value = 0
Next p4
Next p3
Next p2
58
Next p1
Next i
’Identify minimal sets of vanishing Plücker coordinates.
For i = 1 To count - 1
For k = i + 1 To count
j=0
For p1 = 0 To 6
For p2 = p1 + 1 To 7
For p3 = p2 + 1 To 8
For p4 = p3 + 1 To 9
j=j+1
If Cells(1 + i, 5 + j).Value h Cells(1 + k, 5 + j).Value Then GoTo EE
Next p4
Next p3
Next p2
Next p1
j=0
For p1 = 0 To 6
For p2 = p1 + 1 To 7
For p3 = p2 + 1 To 8
For p4 = p3 + 1 To 9
j=j+1
Cells(1 + i, 5 + j).Value = ”x”
Next p4
Next p3
Next p2
Next p1
Exit For
EE: Next k
Next i
’Count the total number of minimal sets found.
59
k=0
For i = 1 To count
If Cells(1 + i, 6).Value hi ”x” Then k = k + 1
Next i
’Generate a message with the total number of minimal sets found.
MsgBox (”I found ” + Str(k) + ” nontrivial sets of vanishing coordinates.”) ’Found
that k=303.
End Sub
Appendix
E
MATLAB code which checks if a
given matrix lies in the general orbit
of GL(4) × SL(5)
In this appendix we provide the MATLAB code, which we used in Chapter 4 in our
proof of stability. In this code cij denote matrix elements of the matrix representing
a given 4-dimensional subspace in P(∧2 V ) and A is the matrix whose determinant
is equal (upto a nonzero scalar multiple) to the relative invariant for the prehomogeneous vector space of GL(4) × SL(5) found in [10]. In Command 2 some matrix
elements cijare assigned zero
0 0 0 ∗ ∗ ∗

0 0 0 0 ∗ ∗
the matrix 
0 0 0 0 0 ∗

0 0 0 0 0 0
values. Herewe work out the example represented by
∗ ∗ ∗ ∗

∗ ∗ ∗ ∗
.
∗ ∗ ∗ ∗

∗ ∗ ∗ ∗
MATLAB code:
Command 1.
syms c10 c11 c12 c13 c14 c15 c16 c17 c18 c19 c20 c21 c22 c23 c24 c25 c26 c27 c28
c29 c30 c31 c32 c33 c34 c35 c36 c37 c38 c39 c40 c41 c42 c43 c44 c45 c46 c47 c48 c49
Command 2.
61
c10=0, c11=0, c12=0, c20=0, c21=0, c22=0, c23=0, c30=0, c31=0, c32=0, c33=0,
c34=0, c40=0, c41=0, c42=0, c43=0, c44=0, c45=0
Command 3.
A=sym([c10 c20 c30 c40 0 0 0 0 0 0 0 0 0 0 0 0 c10 c10 0 0 0 0 0 -c14 0 -c15 0 -c16
0 c11 0 c12 0 c13 0 0 0 0 0 0; 0 0 0 0 c10 c20 c30 c40 0 0 0 0 0 0 0 0 c20 c20 0 0 0
0 0 -c24 0 -c25 0 -c26 0 c21 0 c22 0 c23 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 c10 c20 c30 c40
0 0 0 0 c30 c30 0 0 0 0 0 -c34 0 -c35 0 -c36 0 c31 0 c32 0 c33 0 0 0 0 0 0; 0 0 0 0 0
0 0 0 0 0 0 0 c10 c20 c30 c40 c40 c40 0 0 0 0 0 -c44 0 -c45 0 -c46 0 c41 0 c42 0 c43
0 0 0 0 0 0; c11 c21 c31 c41 0 0 0 0 0 0 0 0 0 0 0 0 c11 0 c11 0 0 c14 0 0 0 -c17 0
-c18 c10 0 0 0 0 0 0 c12 0 c13 0 0 ; 0 0 0 0 c11 c21 c31 c41 0 0 0 0 0 0 0 0 c21 0 c21
0 0 c24 0 0 0 -c27 0 -c28 c20 0 0 0 0 0 0 c22 0 c23 0 0 ; 0 0 0 0 0 0 0 0 c11 c21 c31
c41 0 0 0 0 c31 0 c31 0 0 c34 0 0 0 -c37 0 -c38 c30 0 0 0 0 0 0 c32 0 c33 0 0 ; 0 0 0
0 0 0 0 0 0 0 0 0 c11 c21 c31 c41 c41 0 c41 0 0 c44 0 0 0 -c47 0 -c48 c40 0 0 0 0 0 0
c42 0 c43 0 0 ; c12 c22 c32 c42 0 0 0 0 0 0 0 0 0 0 0 0 c12 0 0 c12 0 c15 0 c17 0 0
0 -c19 0 0 c10 0 0 0 c11 0 0 0 0 c13; 0 0 0 0 c12 c22 c32 c42 0 0 0 0 0 0 0 0 c22 0
0 c22 0 c25 0 c27 0 0 0 -c29 0 0 c20 0 0 0 c21 0 0 0 0 c23; 0 0 0 0 0 0 0 0 c12 c22
c32 c42 0 0 0 0 c32 0 0 c32 0 c35 0 c37 0 0 0 -c39 0 0 c30 0 0 0 c31 0 0 0 0 c33; 0
0 0 0 0 0 0 0 0 0 0 0 c12 c22 c32 c42 c42 0 0 c42 0 c45 0 c47 0 0 0 -c49 0 0 c40 0 0
0 c41 0 0 0 0 c43; c13 c23 c33 c43 0 0 0 0 0 0 0 0 0 0 0 0 0 -c13 -c13 -c13 0 c16 0
c18 0 c19 0 0 0 0 0 0 c10 0 0 0 c11 0 c12 0; 0 0 0 0 c13 c23 c33 c43 0 0 0 0 0 0 0 0
0 -c23 -c23 -c23 0 c26 0 c28 0 c29 0 0 0 0 0 0 c20 0 0 0 c21 0 c22 0; 0 0 0 0 0 0 0 0
c13 c23 c33 c43 0 0 0 0 0 -c33 -c33 -c33 0 c36 0 c38 0 c39 0 0 0 0 0 0 c30 0 0 0 c31 0
c32 0; 0 0 0 0 0 0 0 0 0 0 0 0 c13 c23 c33 c43 0 -c43 -c43 -c43 0 c46 0 c48 0 c49 0 0
0 0 0 0 c40 0 0 0 c41 0 c42 0; c14 c24 c34 c44 0 0 0 0 0 0 0 0 0 0 0 0 0 c14 c14 0 c11
0 -c10 0 0 0 0 0 0 0 0 -c17 0 -c18 0 c15 0 c16 0 0; 0 0 0 0 c14 c24 c34 c44 0 0 0 0 0
0 0 0 0 c24 c24 0 c21 0 -c20 0 0 0 0 0 0 0 0 -c27 0 -c28 0 c25 0 c26 0 0; 0 0 0 0 0 0
0 0 c14 c24 c34 c44 0 0 0 0 0 c34 c34 0 c31 0 -c30 0 0 0 0 0 0 0 0 -c37 0 -c38 0 c35
0 c36 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 c14 c24 c34 c44 0 c44 c44 0 c41 0 -c40 0 0 0 0 0 0
0 0 -c47 0 -c48 0 c45 0 c46 0 0; c15 c25 c35 c45 0 0 0 0 0 0 0 0 0 0 0 0 0 c15 0 c15
c12 0 0 0 -c10 0 0 0 0 c17 0 0 0 -c19 c14 0 0 0 0 c16; 0 0 0 0 c15 c25 c35 c45 0 0 0
0 0 0 0 0 0 c25 0 c25 c22 0 0 0 -c20 0 0 0 0 c27 0 0 0 -c29 c24 0 0 0 0 c26; 0 0 0 0
0 0 0 0 c15 c25 c35 c45 0 0 0 0 0 c35 0 c35 c32 0 0 0 -c30 0 0 0 0 c37 0 0 0 -c39 c34
62
0 0 0 0 c36; 0 0 0 0 0 0 0 0 0 0 0 0 c15 c25 c35 c45 0 c45 0 c45 c42 0 0 0 -c40 0 0 0
0 c47 0 0 0 -c49 c44 0 0 0 0 c46; c16 c26 c36 c46 0 0 0 0 0 0 0 0 0 0 0 0 -c16 0 -c16
-c16 c13 0 0 0 0 0 -c10 0 0 c18 0 c19 0 0 0 0 c14 0 c15 0; 0 0 0 0 c16 c26 c36 c46 0 0
0 0 0 0 0 0 -c26 0 -c26 -c26 c23 0 0 0 0 0 -c20 0 0 c28 0 c29 0 0 0 0 c24 0 c25 0; 0 0
0 0 0 0 0 0 c16 c26 c36 c46 0 0 0 0 -c36 0 -c36 -c36 c33 0 0 0 0 0 -c30 0 0 c38 0 c39
0 0 0 0 c34 0 c35 0; 0 0 0 0 0 0 0 0 0 0 0 0 c16 c26 c36 c46 -c46 0 -c46 -c46 c43 0 0
0 0 0 -c40 0 0 c48 0 c49 0 0 0 0 c44 0 c45 0; c17 c27 c37 c47 0 0 0 0 0 0 0 0 0 0 0 0
0 0 c17 c17 0 0 c12 0 -c11 0 0 0 c15 0 -c14 0 0 0 0 0 0 -c19 0 c18; 0 0 0 0 c17 c27
c37 c47 0 0 0 0 0 0 0 0 0 0 c27 c27 0 0 c22 0 -c21 0 0 0 c25 0 -c24 0 0 0 0 0 0 -c29
0 c28; 0 0 0 0 0 0 0 0 c17 c27 c37 c47 0 0 0 0 0 0 c37 c37 0 0 c32 0 -c31 0 0 0 c35 0
-c34 0 0 0 0 0 0 -c39 0 c38; 0 0 0 0 0 0 0 0 0 0 0 0 c17 c27 c37 c47 0 0 c47 c47 0 0
c42 0 -c41 0 0 0 c45 0 -c44 0 0 0 0 0 0 -c49 0 c48; c18 c28 c38 c48 0 0 0 0 0 0 0 0 0 0
0 0 -c18 -c18 0 -c18 0 0 c13 0 0 0 -c11 0 c16 0 0 0 -c14 0 0 c19 0 0 c17 0; 0 0 0 0 c18
c28 c38 c48 0 0 0 0 0 0 0 0 -c28 -c28 0 -c28 0 0 c23 0 0 0 -c21 0 c26 0 0 0 -c24 0 0
c29 0 0 c27 0; 0 0 0 0 0 0 0 0 c18 c28 c38 c48 0 0 0 0 -c38 -c38 0 -c38 0 0 c33 0 0 0
-c31 0 c36 0 0 0 -c34 0 0 c39 0 0 c37 0; 0 0 0 0 0 0 0 0 0 0 0 0 c18 c28 c38 c48 -c48
-c48 0 -c48 0 0 c43 0 0 0 -c41 0 c46 0 0 0 -c44 0 0 c49 0 0 c47 0; c19 c29 c39 c49 0 0
0 0 0 0 0 0 0 0 0 0 -c19 -c19 -c19 0 0 0 0 0 c13 0 -c12 0 0 0 c16 0 -c15 0 c18 0 -c17
0 0 0; 0 0 0 0 c19 c29 c39 c49 0 0 0 0 0 0 0 0 -c29 -c29 -c29 0 0 0 0 0 c23 0 -c22 0
0 0 c26 0 -c25 0 c28 0 -c27 0 0 0; 0 0 0 0 0 0 0 0 c19 c29 c39 c49 0 0 0 0 -c39 -c39
-c39 0 0 0 0 0 c33 0 -c32 0 0 0 c36 0 -c35 0 c38 0 -c37 0 0 0; 0 0 0 0 0 0 0 0 0 0 0
0 c19 c29 c39 c49 -c49 -c49 -c49 0 0 0 0 0 c43 0 -c42 0 0 0 c46 0 -c45 0 c48 0 -c47 0 0 0])
Command 4.
rank(A)
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Vita
Evgeny Mayanskiy
The Pennsylvania State University
Mathematics Departmant
109 McAllister Building
University Park, PA 16802-6401
Citizenship: Russia, US F-1 Visa
Current address:
411 Waupelani Drive Apt C-033
State College, PA 16801
Email: [email protected]
Research Interests
Algebraic Geometry (Advisor: Yuri Zarhin)
Education
2007 - 2013 (expected) PhD Pennsylvania State University, Mathematics
2004 M.A. Moscow Institute of Physics and Technology
2002 B.S. Moscow Institute of Physics and Technology
Publications
(all of them are ArXiv preprints)
Chow groups of moduli spaces of rank 2 vector bundles on curves with determinant of odd degree, arXiv:1111.2986 13 Nov 2011
Intersection lattices of cubic fourfolds, arXiv:1112.0806 4 Dec 2011
Stacky analogs of elliptic surfaces I. Orbifold elliptic fibrations, in preparation
Tetrahedral quartics in Projective Space, arXiv:1206.5903 26 Jun 2012
Hermitian forms of K3 type, arXiv:1210.0189 30 Sep 2012
Endomorphism algebras of Kuga-Satake varieties, arXiv:1210.0190v2 2 Nov 2012
Academic Honors
2007-2008 Pennsylvania State University Graduate Fellowship