Introduction
The idea of the algorithm
The implementation
Experimental results
AN EFFICIENT IMPLEMENTATION OF THE
ALGORITHM COMPUTING THE BOREL-FIXED
POINTS OF A HILBERT SCHEME
Paolo Lella
University of Turin
ISSAC 2012
Grenoble, France, 22-25 July 2012
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
1 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
The Hilbert scheme
Given an admissible Hilbert polynomial p(t) in the projective space Pn , the Hilbert
scheme Hilbnp(t) parametrizes all subschemes and all flat families of subschemes of
Pn with Hilbert polynomial p(t).
The Hilbert scheme is an important object in deformation theory as the Hilbert
polynomial determines lots of geometrical properties.
For instance, all the curves of degree d and genus g have Hilbert polynomial
p(t) = dt + 1 − g .
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
2 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Borel-fixed points
Definition
A Borel-fixed point of the Hilbert scheme Hilbnp(t) is a point corresponding to a
subscheme X ⊂ Pn with Hilbert polynomial p(t) defined by a Borel-fixed ideal
I ⊂ K[x0 , . . . , xn ], i.e. X = Proj K[x0 , . . . , xn ]/I .
Definition
A homogeneous ideal I ⊂ K[x0 , . . . , xn ] is called Borel-fixed if it is fixed by the
action of the Borel subgroup of GL(n + 1), that is the group of upper triangular
matrices.
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
3 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
The importance of Borel-fixed ideals
Borel-fixed ideals are much studied because in the case of a base field K of
characteristic 0, they have a nice combinatorial description.
Proposition
If char K = 0, a homogeneous ideal I ⊂ K[x0 , . . . , xn ] is Borel-fixed if and only if
1
I is a monomial ideal;
2
for any x α ∈ I , then x β =
Paolo Lella (University of Turin)
xj α
x ∈ I , ∀ xj > xi and xi | x α .
xi
Borel-fixed points of a Hilbert scheme
July 23, 2012
4 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
The importance of Borel-fixed points
Each component and each intersection of components
of the Hilbert scheme contains at least one Borel-fixed point.
Roughly, we can say that they are distributed all over the Hilbert scheme.
Borel-fixed points have been used
in the proof of the connectedness of Hilbnp(t) (Hartshorne 1966,
Peeva-Stillman 2005, L. 2010)
in the computation of global equations of low degree defining Hilbnp(t)
(Brachat-L.-Mourrain-Roggero 2011)
in the construction of an open cover of Hilbnp(t) (Bertone-L.-Roggero 2011)
...
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
5 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
A first algorithm
It has been natural to ask if it was possible to determine all the Borel-fixed points
of a Hilbert scheme, i.e. given a polynomial ring K[x0 , . . . , xn ] and a Hilbert
polynomial p(t) if it was possible to compute all the (saturated) Borel-fixed ideals
of K[x0 , . . . , xn ] defining schemes with Hilbert polynomial p(t).
Cioffi-L.-Marinari-Roggero 2011
We gave a positive answer to the previous question exhibiting an algorithm
computing all such Borel-fixed ideals.
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
6 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Notation
K will be an algebraically closed field of characteristic 0.
We consider the polynomial ring in n + 1 variables K[x0 , . . . , xn ] assuming
x0 < · · · < xn .
For any monomial x α = x0α0 · · · xnαn we denote min x α = min{i | αi > 0} and
max x α = max{j | αj > 0}.
Let p(t) be a Hilbert polynomial. We denote ∆0 p(t) = p(t) and
∆i p(t) = ∆i−1 p(t) − ∆i−1 p(t − 1).
Paolo Lella (University of Turin)
∆deg p(t)+1 p(t) = 0.
Borel-fixed points of a Hilbert scheme
July 23, 2012
7 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
The Borel order
The transitive closure of the order
xα <
xi+1 α
x , ∀ xi | x α
xi
on the monomials of K[x0 , . . . , xn ] of fixed degree r is a partial order. We called it
Borel order and denote it ≤B .
By the definition, the monomials of fixed degree r of a Borel-fixed ideal I
represent a filter for the Borel order on the monomials of K[x0 , . . . , xn ]r , that is
xi+1
the set of monomials in Ir is closed under the multiplication by
.
xi
We call Borel set such a set.
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
8 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
An example: K[x0 , x1 , x2 ]4
x24
x2
x1
x23 x1
x23 x0
x22 x12
x2 x13
x14
x22 x1 x0
x2 x12 x0
x1
x0
3
x1 x0
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
9 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
An example: K[x0 , x1 , x2 ]4
x24
x23 x1
x22 x12
x2 x13
x14
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
9 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Minimal monomials
x24
x23 x1
x22 x12
x2 x13
x14
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
10 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
A suitable degree
Definition
Let p(t) be a Hilbert polynomial. The number of terms in the unique
decomposition
t + a1
t + a2 − 1
t + ar − r + 1
p(t) =
+
+ ... +
,
a1
a2
ar
a1 > . . . > ar
is called Gotzmann number of p(t).
Gotzmann’s Regularity Theorem
Let p(t) be a Hilbert polynomial with Gotzmann number r . Any saturated
Borel-fixed ideal I ⊂ K[x0 , . . . , xn ] defining a scheme with Hilbert polynomial p(t)
is generated in degree 6 r , that is
hIr i = I>r and dim Ir = dim K[x0 , . . . , xn ]r − p(r ).
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
11 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Hence we have the injective map






 Borel set of K[x , . . . , x ] containing 
Borel-fixed ideal defining
0
n r
,→
 a Borel-fixed point of Hilbn 
 dim K[x , . . . , x ] − p(r ) monomials 
0
n r
p(t)
Which is the image of such a map?
Theorem (L. 2012)


Borel-fixed ideal defining
 a Borel-fixed point of Hilbn
p(t)
Paolo Lella (University of Turin)



Borel set B of K[x0 , . . . , xn ]r s.t.







∀ i, B ∩ K[xi , . . . , xn ] has
1:1
←→



dim K[xi , . . . , xn ]r − ∆i p(r )





monomials
Borel-fixed points of a Hilbert scheme
July 23, 2012















12 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Example
Consider the projective plane P2 and the Hilbert polynomial p(t) = t + 4.
The Gotzmann number of p(t) is 4:
t +1
t −1
t −2
t −3
p(t) =
+
+
+
1
0
0
0


Borel-fixed ideal defining
 a Borel-fixed point of Hilb2
t+4
Paolo Lella (University of Turin)






Borel
set
B
of
K[x
,
x
,
x
]
s.t.
0 1 2 4












|B ∩ K[x0 , x1 , x2 ]4 | = 7
1:1
←→





|B ∩ K[x1 , x2 ]4 | = 4










|B ∩ K[x2 ]4 | = 1
Borel-fixed points of a Hilbert scheme
July 23, 2012
13 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
The main idea
The main idea is to use a recursive algorithm, i.e. to determine






Borel
set
B
of
K[x
,
.
.
.
,
x
]
s.t.
0
n r










∀ i, B ∩ K[xi , . . . , xn ] has




dim K[xi , . . . , xn ]r − ∆i p(r )










monomials
knowing


e of K[x1 , . . . , xn ]r s.t.

Borel set B




 ∀ i > 1, B
e ∩ K[xi , . . . , xn ] has


dim K[xi , . . . , xn ]r − ∆i p(r )





monomials
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme















July 23, 2012
14 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
A slim structure
To describe a Borel set B ⊂ K[xi , . . . , xn ] we use a very slim structure. We store
a single information:
the set of minimal monomials of the Borel set.
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
15 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Example of execution. P2 and p(t) = t + 4
To determine
we need to know






Borel
set
B
of
K[x
,
x
,
x
]
s.t.
0 1 2 4










|B ∩ K[x0 , x1 , x2 ]4 | = 7




|B ∩ K[x1 , x2 ]4 | = 4










|B ∩ K[x2 ]4 | = 1





Borel set B of K[x1 , x2 ]4 s.t. 




|B ∩ K[x1 , x2 ]4 | = 4








|B ∩ K[x ] | = 1
2 4
and


 Borel set B of K[x ] s.t. 
2 4


|B ∩ K[x ] | = 1
2 4
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
16 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
The base of the recursive process
B ⊂ K[x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1
x24
x23 x1
x22 x12
x2 x13
x14
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
17 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
The base of the recursive process
B ⊂ K[x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1
x24
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x24
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
17 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
First recursive call
B ⊂ K[x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1 and |B ∩ K[x1 , x2 ]4 | = 4
x24
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x24
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
18 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
First recursive call
B ⊂ K[x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1 and |B ∩ K[x1 , x2 ]4 | = 4
OLD
Minimal
monomials:
NEW
x24
x24
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x14
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
18 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Removing monomials
B ⊂ K[x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1 and |B ∩ K[x1 , x2 ]4 | = 4
x24
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x14
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
19 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Removing monomials
B ⊂ K[x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1 and |B ∩ K[x1 , x2 ]4 | = 4
OLD
Minimal
monomials:
NEW
x24
x14
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x2 x13
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
19 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Second recursive call
B ⊂ K[x0 , x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1,
|B ∩ K[x1 , x2 ]4 | = 4 and |B ∩ K[x0 , x1 , x2 ]4 | = 7
x24
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x2 x13
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
20 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Second recursive call
B ⊂ K[x0 , x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1,
|B ∩ K[x1 , x2 ]4 | = 4 and |B ∩ K[x0 , x1 , x2 ]4 | = 7
OLD
Minimal
monomials:
NEW
x24
x2 x13
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x2 x03
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
20 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Removing monomials
B ⊂ K[x0 , x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1,
|B ∩ K[x1 , x2 ]4 | = 4 and |B ∩ K[x0 , x1 , x2 ]4 | = 7
x24
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x2 x03
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
21 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Removing monomials
B ⊂ K[x0 , x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1,
|B ∩ K[x1 , x2 ]4 | = 4 and |B ∩ K[x0 , x1 , x2 ]4 | = 7
OLD
Minimal
monomials:
NEW
x24
x2 x03
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x2 x1 x02
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
21 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Removing monomials
B ⊂ K[x0 , x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1,
|B ∩ K[x1 , x2 ]4 | = 4 and |B ∩ K[x0 , x1 , x2 ]4 | = 7
x24
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x2 x1 x02
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
21 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Removing monomials
B ⊂ K[x0 , x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1,
|B ∩ K[x1 , x2 ]4 | = 4 and |B ∩ K[x0 , x1 , x2 ]4 | = 7
OLD
Minimal
monomials:
NEW
x24
x2 x1 x02
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x22 x02
x2 x12 x0
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
21 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Removing monomials
B ⊂ K[x0 , x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1,
|B ∩ K[x1 , x2 ]4 | = 4 and |B ∩ K[x0 , x1 , x2 ]4 | = 7
x24
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x22 x02
x2 x12 x0
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
21 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
First Borel-fixed ideal: (x22 , x2 x13 )
B ⊂ K[x0 , x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1,
|B ∩ K[x1 , x2 ]4 | = 4 and |B ∩ K[x0 , x1 , x2 ]4 | = 7
x24
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x22 x02
x2 x12 x0
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
22 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
First Borel-fixed ideal: (x22 , x2 x13 )
B ⊂ K[x0 , x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1,
|B ∩ K[x1 , x2 ]4 | = 4 and |B ∩ K[x0 , x1 , x2 ]4 | = 7
OLD
Minimal
monomials:
NEW
x24
x22 x02
x2 x12 x0
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x22 x02
x2 x13
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
22 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Second Borel-fixed ideal: (x23 , x22 x1 , x2 x12 )
B ⊂ K[x0 , x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1,
|B ∩ K[x1 , x2 ]4 | = 4 and |B ∩ K[x0 , x1 , x2 ]4 | = 7
x24
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x22 x02
x2 x12 x0
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
23 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Second Borel-fixed ideal: (x23 , x22 x1 , x2 x12 )
B ⊂ K[x0 , x1 , x2 ]4 s.t. |B ∩ K[x2 ]4 | = 1,
|B ∩ K[x1 , x2 ]4 | = 4 and |B ∩ K[x0 , x1 , x2 ]4 | = 7
OLD
Minimal
monomials:
NEW
x24
x23 x1
x22 x12
x2 x13
x14
Minimal
monomials:
x23 x0
x22 x1 x0
x2 x12 x0
x13 x0
x2 x12 x0
x22 x02
x2 x1 x02
x12 x02
x2 x03
x1 x03
x04
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
23 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Points (Hilbert polynomials of degree 0)
Hilbnp(t)
n=5
n = 10
n = 15
n = 20
p(t) = 5
5 (0.101)
5 (0.025)
5 (0.017)
5 (0.021)
p(t) = 10
42 (0.062)
50 (0.064)
50 (0.119)
50 (0.048)
p(t) = 15
287 (0.079)
417 (0.225)
425 (0.298)
425 (0.401)
p(t) = 20
1732 (0.341)
3130 (1.595)
3263 (2.735)
3271 (3.870)
p(t) = 25
9501 (2.094)
21616 (13.595)
23158 (24.497)
23291 (33.303)
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
24 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Curves (Hilbert polynomials of degree 1)
Hilbnp(t)
n=5
n = 10
n = 15
n = 20
5t + 1
89 (0.117)
98 (0.159)
98 (0.067)
98 (0.0621)
5t + 7
3028 (0.502)
4560 (1.480)
4587 (2.312)
4587 (3.290)
8t − 6
4171 (0.987)
6741 (2.623)
6837 (4.138)
6838 (5.852)
8t − 3
17334 (3.008)
32073 (14.128)
32848 (22.960)
32868 (32.300)
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
25 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
Surfaces (Hilbert polynomials of degree 2)
Hilbnp(t)
n=5
n = 10
n = 15
n = 20
2
2t + 8t − 46
34 (0.312)
38 (0.189)
38 (0.304)
38 (0.516)
2
2t + 8t − 42
481 (0.103)
670 (0.338)
671 (0.558)
671 (0.883)
4t 2 − 12t + 10
631 (0.147)
856 (0.280)
857 (0.377)
857 (0.561)
4t 2 − 12t + 14
6394 (0.953)
10986 (3.909)
11082 (6.007)
11082 (8.588)
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
26 / 27
Introduction
The idea of the algorithm
The implementation
Experimental results
My web page
The algorithm is available at
www.personalweb.unito.it/paolo.lella/HSC/
Paolo Lella (University of Turin)
Borel-fixed points of a Hilbert scheme
July 23, 2012
27 / 27