Secant Varieties, Symbolic Powers,
Statistical Models
Seth Sullivant
North Carolina State University
November 19, 2012
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
1 / 27
Joins and Secant Varieties
V
v
Given V , W ⊆ Pn−1 , the join is
V ∗ W :=
[
w
hv , wi ⊆ Pn−1
W
v ∈V ,w∈W
Given V ⊂ Pn−1 , the r -th secant variety of V is given by
Sec1 (V ) = V , and
Secr (V ) := V ∗ Secr −1 (V ) =
[
< v1 , v2 , . . . , vr > ⊆ Pn−1
v1 ,...,vr ∈V
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
2 / 27
Joins and Secant Ideals
Definition
Let I, J ⊆ K[x1 , . . . , xn ] homogeneous ideals. The join of I and J is
I ∗ J := (I(y ) + J(z) + hxi − yi − zi : i = 1, . . . , ni) ∩ K[x1 , . . . , xn ].
i.e. f ∈ I ∗ J ⇐⇒ f (y + z) ∈ I(y ) + J(z).
Definition
The r -th secant ideal I ⊆ K[x1 , . . . , xn ] is defined by
1
I {1} = I
2
I {r } = I ∗ I {r −1} , r > 1.
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
3 / 27
Matrices of Bounded Rank
Consider K[X ] := K[xij : 1 ≤ i ≤ m,
1 ≤ j ≤ n]
Let I be the ideal of 2 × 2 minors of generic matrix


x11 x12 · · · x1n

..
.. 
..
X =  ...
.
. 
.
xm1 xm2 · · ·
xmn
V (I) ⊆ Pmn−1 is the set of rank 1 matrices. (aka Seg(Pm−1 × Pn−1 ))
Secr (V (I)) is the set of all matrices of rank ≤ r .
I {r } = I(Secr (V (I))) is the ideal of (r + 1) × (r + 1) minors of X .
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
4 / 27
When is the Secant Ideal “Nice”?
Question
For which ideals I does I {2} = h0i?
i.e. When is V (I){2} = Pn−1 ?
Theorem (Zak 1994)
If V (I) is a linearly nondegenerate, smooth irreducible variety of
codimension ≤ 13 n + 1, then I {2} = h0i unless V (I) is one of
ν2 (P2 ),
Seg(P2 × P2 ),
Gr (2, 6),
or the Cartan variety.
Question
For which ideals I is I {r } generated by polynomials of degree r + 1?
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
5 / 27
Symbolic Powers
Definition
The r th symbolic power of I ⊂ R is the ideal
I (r ) = (RI−1 · I r ) ∩ R
where RI is the complement of the union of the minimal primes of I.
If P is a prime ideal, P (r ) is the P-primary component of P r .
Theorem ((Special case of) Zariski-Nagata)
If I ⊆ C[x1 , . . . , xn ] is radical then
*
+
n
|a| f
X
∂
I (r ) = I <r > := f |
∈ I for all a ∈ Nn with
ai ≤ r − 1 .
∂x a
i=1
Symbolic power gives equations vanishing to high order on V (I).
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
6 / 27
Ideal of 2 × 2 minors
Consider K[X ] := K[xij : 1 ≤ i ≤ m,
1 ≤ j ≤ n]
Let I be the ideal of 2 × 2 minors of generic matrix X .
The partial derivatives of minors of a generic matrix are
themselves minors of a generic matrix. e.g.
x11 x12 x13 x22 x23 ∂ x21 x22 x23 = x32 x33 ∂x11 x31 x32 x33
I (2) generated by 3 × 3 minors and products of 2 2 × 2 minors of X .
Note, if m = 2, then I (2) = I 2 .
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
7 / 27
When are the Symbolic Powers “Nice”? I
Definition
A radical ideal I is normally torsion free if
I (r ) = I r
for all r ≥ 1.
Complete intersections, Maximal minors
Theorem (Ein-Lazarsfeld-Smith, Hochster-Huneke)
I (codim(I)r ) ⊆ I r for all r in an equal characteristic regular ring.
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
8 / 27
Secant Ideals and Symbolic Powers
Proposition
I <r > = I ∗ hx1 , . . . , xn ir .
For I ⊆ C[x1 , . . . , xn ],
Proposition (Catalano-Johnson)
If I ⊆ C[x1 , . . . , xn ], homogeneous and I ⊆ hx1 , . . . , xn i2 , then
I {r } ⊆ I <r > .
Proof.
I {r }
=
I ∗ I {r −1}
⊆
I ∗ hx1 , . . . , xn ir
=
I <r > .
Proposition (Landsberg-Manivel, Sidman-Sullivant)
If I ⊆ C[x1 , . . . , xn ], homogeneous and I ⊆ hx1 , . . . , xn id , then
<(d−1)(r −1)+1>
I(d−1)r +1
Seth Sullivant (NCSU)
{r }
= I(d−1)r +1
Secant Varieties, etc.
November 19, 2012
9 / 27
When are the Symbolic Powers “nice”? II
Catalano-Johnson says I {r } ⊆ I <r > .
Product rule gives I <r −i> I <i> ⊆ I <r > , i = 1, . . . , r − 1.
Putting together gives
X
I {λ1 } · · · I {λl } ⊆ I <r >
λ`r
where the sum is over all partitions of r .
Definition
An ideal I ⊆ C[x1 , . . . , xn ] is differentially perfect if and only if
X
for all r ≥ 1.
I <r > =
I {λ1 } · · · I {λl }
λ`r
Normally torsion free ideal, determinantal and Pfaffian ideals
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
10 / 27
Secants and Symbolic Powers of Monomial Ideals
Definition
Let G be an undirected graph, with vertex set [n] = {1, 2, . . . , n}. The
edge ideal of G is
I(G) = hxi xj : ij ∈ E(G)i.
2
1
6
3
4
I(G) =
hx1 x2 , x2 x3 , x3 x4 , x4 x5 , x5 x6 , x1 x6 , x2 x4 i
5
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
11 / 27
Secants of Edge Ideals and Colorings
Given a graph G and subset of vertices V , GV is the induced
subgraph of G with vertex set V .
The chromatic number χ(G) is the smallest number of colors to
properly color the vertices of a graph.
Theorem (Sturmfels-Sullivant)
The secant ideals of an edge ideal I(G) are generated by
Y
I(G){r } = h xi : χ(GV ) > r i.
i∈V
Minimal generators of I(G){r } are minimal obstructions to r -coloring G.
2
1
6
3
4
I(G){2} = hx2 x3 x4 ,
x1 x2 x4 x5 x6 i
5
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
12 / 27
“Nice” Secant Ideals of Edge Ideals
Proposition
I(G){2} = h0i if and only if G is a bipartite graph.
Proposition
I(G){r } is generated in degree ≤ r + 1 for all r if and only if G is a
perfect graph.
2
1
6
2
1
3
6
4
5
3
4
5
Not perfect
Perfect
I(G){2} = hx2 x3 x4 , x1 x2 x4 x5 x6 i
I(G){2} = hx1 x2 x5 , x1 x5 x6 , x2 x3 x4 , x2 x4 x5 i
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
13 / 27
Symbolic Powers of Edge Ideals and Coverings
2
1
6
3
The covering number τ (G) is the size of
the smallest vertex cover of G.
4
5
a = (1, 2, 1, 1, 0, 0)
2
1
3
2
For a vector a ∈ Nn the parallelization Ga
is the graph with each vertex i replicated
ai times.
4
Theorem
I(G)<r > = hx a : τ (Ga ) ≥ r i
Generators of symbolic powers are minimal obstructions to covering.
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
14 / 27
“Nice” Symbolic Powers of Edge Ideals
Theorem (Simis-Vasconcelos-Villarreal)
I(G)<r > = I(G)r for all r if and only if G is a bipartite graph.
Theorem (Villarreal, Sullivant)
I(G) is differentially perfect, i.e.
X
I(G)<r > =
I(G){λ1 } · · · I(G){λl }
λ`r
for all r if and only if G is a perfect graph.
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
15 / 27
Statistical Models
Definition
A statistical model is a set of probability distributions or densities.
Independence model for two discrete random variables:




X
m×n
MX ⊥
=
P
=
(p
)
∈
R
:
p
=
1,
p
≥
0,
rank
P
=
1
ij
ij
ij
⊥Y 

i,j
(Zariski closure of ) many statistical models have structure of
algebraic varieties
Many of those varieties are secant varieties or joins
Mixture models
Factor analysis
Can use that algebraic structure to answer statistical questions.
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
16 / 27
Join vs. Mixture
V
v
Join
w
W
V
v
Mixture
w
W
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
17 / 27
Identifiability of Phylogenetic Mixture Models
Theorem (Rhodes-Sullivant)
The unrooted tree and numerical parameters in a r -class, same tree
phylogenetic mixture model on n-leaf trivalent trees are
generically identifiable, if r < 4dn/4e .
Identifiability for Secant Varieties: Does a generic point on the r th
secant variety Secr (V ) lie on a unique secant r − 1 plane to V ?
Proof uses:
Relation to identifiability problem for Sec 4r (Seg(Pa × Pb × Pc ))
{r }
Knowledge of generators of the secant ideals IT
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
18 / 27
Factor Analysis
A centered Gaussian random vector X ∈ Rn has density function
1
1 T −1
f (x) =
x
Σ
x
exp
−
2
(2π)n/2 |Σ|1/2
with Σ ∈ PDn ⊂ Rn(n+1)/2 .
Set of all centered Gaussian r.v.’s parametrized by PDn .
A Gaussian statistical model is a subset Θ ⊂ PDn .
Definition (Factor Analysis Model)
Θr ,n = {Ψ + ΛΛT | Ψ > 0 diagonal , Λ ∈ Rr ×n } ⊂ PDn
Diagonal plus rank ≤ r PSD matrix.
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
19 / 27
Factor Analysis
Model assumptions: X1 , . . . , Xn are observable covariates. Y1 , . . . , Yr
are fewer unobservable (hidden) factors, which explain the correlation
among X1 , . . . , Xn .
For all i, j, Xi ⊥
⊥Xj |Y1 , . . . , Yr
Y1
For all i, j Yi ⊥
⊥Yj
Y2
Example (g-theory)
Xi are test scores, and Yj are
underlying “types of intelligence”.
Seth Sullivant (NCSU)
Secant Varieties, etc.
X1
X2
X3
X4
X5
November 19, 2012
20 / 27
Model Selection: How Many Factors? What is r ?
Statistical Approach
Mathematical Problem
Wald-Type Test
Find generators of I(Θr ,n )
Likelihood Ratio
Test
Compute tangent cone TCp (Θr ,n ) at
p ∈ Θr −1,n .
Information
Criteria (AIC,
BIC, WAIC)
Determine the Real Log-Canonical
Threshold of singular fibers of Θr ,n .
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
21 / 27
The Second Hypersimplex
Observation
Θr ,n = Secr (Θ1,n )
Θ1,n is a toric variety called the second hypersimplex.
φ : P2n−1 99K Pn(n+1)/2−1
λi λj
if i 6= j
φij (ψ, λ) =
2
ψi + λi if i = j
Definition
I1,n = I(im φ) toric ideal of second hypersimplex
{r }
Ir ,n = I1,n = vanishing ideal of r -factor model
Quadratic Gröbner basis of I1,n computed by
De Loera-Sturmfels-Thomas
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
22 / 27
Theorem (Sullivant)
{2}
The secant ideal I2,n = I1,n has a Gröbner basis, with squarefree initial
terms, consisting of polynomials of all odd degrees between 3 and n,
with respect to a circular term order.
Proof idea:
{2}
Degenerative strategy: Hope that in≺ (I1,n ) = (in≺ (I1,n )){2}
Lemma (Simis-Ulrich)
For any I and any term order ≺
in≺ (I {r } ) ⊆ (in≺ (I)){r }
Just need to construct polynomials in I {r } whose initial terms
generate (in≺ (I)){r } .
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
23 / 27
Initial Ideal Structure and Symbolic Powers
The initial ideal in≺ (I1,n ) is an edge ideal.
The graph Gn such that in≺ (I1,n ) = I(Gn ) is the non-crossing pair
graph of the cyclically embedded complete graph Kn . [De
Loera-Sturmfels-Thomas]
in≺ (I1,n ){2} is generated by the odd cycles in Gn .
Construct polynomials in I2,n having those cycles as initial terms.
σ12 σ15 σ23 σ34 σ45 −σ12 σ13 σ25 σ34 σ45 −σ12 σ14 σ23 σ35 σ45 +σ12 σ14 σ25 σ34 σ35 +σ12 σ13 σ24 σ35 σ45 −σ12 σ15 σ24 σ34 σ35
+σ13 σ14 σ23 σ25 σ45 −σ13 σ14 σ24 σ25 σ35 −σ13 σ15 σ23 σ24 σ45 +σ13 σ15 σ24 σ25 σ34 −σ14 σ15 σ23 σ25 σ34 +σ14 σ15 σ23 σ24 σ35
{2}
Prove that these generalized pentads are in I1,n using symbolic
powers.
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
24 / 27
Summary
Secant varieties and symbolic powers are intimately related.
Symbolic powers shed light on equations for secant varieties.
In special cases, symbolic powers’ structure complete determined
by secant ideals.
Nice combinatorial and computational structure.
Secant varieties make frequent appearance in statistics.
Mixture models
Factor analysis model
Theoretical advances on secant varieties lead to advances in
statistics.
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
25 / 27
References
M. Catalano-Johnson. The homogeneous ideals of higher secant varieties. J. Pure Appl. Algebra 158 (2001), no. 2-3,
123–129.
J. de Loera, B. Sturmfels, R. Thomas. Gröbner bases and triangulations of the second hypersimplex. Combinatorica 15
(1995), no. 3, 409–424.
L. Ein, R. Lazarsfeld, K. Smith. Uniform bounds and symbolic powers on smooth varieties. Invent. Math. 144 (2001), no. 2,
241–252.
D. Eisenbud and M. Hochster. A Nullstellensatz with nilpotents and Zariski’s main lemma on holomorphic functions. J.
Algebra 58 (1979), no. 1, 157–161.
M. Haiman. Commutative algebra of n points in the plane. With an appendix by Ezra Miller. Math. Sci. Res. Inst. Publ., 51,
Trends in commutative algebra, 153–180, Cambridge Univ. Press, Cambridge, 2004.
M. Hochster, C. Huneke. Comparison of symbolic and ordinary powers of ideals. Invent. Math. 147 (2002), no. 2, 349–369.
J. Landsberg, L. Manivel. On the projective geometry of rational homogeneous varieties. Comment. Math. Helv. 78 (2003),
no. 1, 65–100.
J. A. Rhodes and S. Sullivant, Identifiability of large phylogenetic mixture models. Bull. Math. Biol. 74 (2012), no. 1,
212–231.
J. Sidman and S. Sullivant. Prolongations and computational algebra. Canadian Journal of Mathematics 61 no. 4 (2009)
930–949
A. Simis, B. Ulrich. On the ideal of an embedded join. J. Algebra 226 (2000), no. 1, 1–14.
Seth Sullivant (NCSU)
Secant Varieties, etc.
November 19, 2012
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A. Simis, W. Vasconcelos. R. Villarreal. On the ideal theory of graphs. J. Algebra 167 (1994), no. 2, 389–416.
B. Sturmfels and S. Sullivant. Combinatorial secant varieties. Quarterly Journal of Pure and Applied Mathematics 2 (2006)
285–309
S. Sullivant Combinatorial symbolic powers. J. Algebra 319 (2008), no. 1, 115–142.
S. Sullivant. A Gröbner basis for the secant ideal of the second hypersimplex. Journal of Commutative Algebra 1 no.2
(2009)
R. H. Villarreal, Rees algebras and polyhedral cones of ideals of vertex covers of perfect graphs, J. Algebraic Combin. 27
(2008), no. 3, 293–305.
F. L. Zak. Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, 127, AMS Press, 1993.
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Secant Varieties, etc.
November 19, 2012
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