Posts tagged ‘geometry’

Picard lattices of Fano threefolds

The updated script for computing Picard lattices of Fano threefolds: now it works and, even better, computes all five principal invariants of the smoothing!

Picard lattices of ambiguous nodal toric Fano threefolds

Let X be a nodal toric Fano threefold (recall that in toric world terminal Gorenstein singularities of Fano threefolds are simply ordinary double points aka nodes (xy=zt) \subset \CC^4 = \mathrm{Spec } \CC[x,y,z,t] ).

Given a terminal Gorenstein toric Fano threefold X,

this script do the following:

1. Compute Picard lattice Pic(X)
2. Then compute (self)intersection theory on this lattice.
This part is done in 3 steps:
a. pick a small crepant resolution \pi : \hat{X} \rightarrow X
b. compute intersection theory of smooth toric manifold \hat{X},
c. restrict intersection theory from Pic \hat{X} to \pi^* Pic X \cong Pic X.

3. Threefold X has a unique deformation class of smoothing by Fano threefold Y
We also compute the principal invariants of Y: Betti numbers, degree, Lefschetz discriminant and Fano index

The main procedure is called Picard(toric)
The input is a 3-component vector toric=[description, vertices, faces]
description is a verbal description of variety X (not used for computations)
vertices is a matrix of vertices of the fan polytope Delta(X)
faces is a transposed matrix of faces (vertices of the moment polytope)

The output is a 2-component vector o = [lattice, invariants]
where lattice is 3-component vector [cubic, M, class]
cubic is homogenous cubic polynomial of ‘rk Pic(X)’ variables (self-intersection pairing)
class is the expression of the first Chern class in terms of generators of Picard group
M is the matrix of the Lefschetz pairing (a,b) \to \int_{[X]} a \cup b \cup c_1(X)
and invariants is 5-component vector [rho,deg,b,d,r] of the principal invariants
rho is Picard number i.e. second Betti number of X
deg is the anticanonical degree \int_{[X]} c_1(X)^3
b is the half of third Betti number of the smoothing Y
d is the Lefschetz discriminant (i.e. determinant of matrix M)
r is the Fano index (i.e. divisibility of c_1(X) in H^2(X,\ZZ))

Quantum Lefshetz for non-split bundle via “abelian/non-abelian correspondence”.

Bumsig Kim explained me how their beautiful theory provides a tool for computing J-series of V_{22} and many other Fano threefolds.

The computation for V_{22} can be reproduced by the following pari/gp code (I omit checking that mirror map is almost trivial):

o = O(t^(N+1))
h(n) = sum(k=1,n,1/k)
hh(n) = sum(k=1,n,sum(l=k+1,n,1/k/l))
gg(x, p) = (x!*(1+h(x)*p+hh(x)*p^2+O(u^4)))
simplemirrormap(F) = F * exp(-polcoeff(F,1,t)*t)
reg(F) = sum(n=0,N,t^n*polcoeff(F,n,t)*n!)+O(t^(N+1))
period(F) = reg(simplemirrormap(F))

gp > v22 = period( polcoeff( sum(a=0,N,sum(b=0,N,sum(c=0,N,o+ t^(a+b+c)* ( gg(a+b,u*(A+B)) * gg(a+c,u*(A+C)) * gg(b+c,u*(B+C)) )^3 / (gg(a,u*A) * gg(b,u*B) * gg(c,u*C) )^7 *(c-b+u*(C-B))*(c-a+u*(C-A))*(b-a+u*(B-A)) ))) + O(u^4) , 3,u) /(C-B)/(C-A)/(B-A) + o )

%2 = 1 + 12*t^2 + 60*t^3 + 636*t^4 + 5760*t^5 + 58620*t^6 + 604800*t^7 + 6447420*t^8 + O(t^9)

Indeed, period sequence 17.
Note that gg(x,p) = \frac{\Gamma(1 + x + \frac{p}{z})}{\Gamma(1 + \frac{p}{z})} + o(\frac{1}{z^3}) is the familiar Gamma-factor with u=\frac{1}{z}.


So, how does it works?

Consider 3-dimensional vector space U = \CC^3 with a fixed base, 7-dimensional vector space V = \CC^7, and space M = \CC^{21} of 3×7 matrices M = Hom(U,V) =Hom(\CC^3, \CC^7)

Group G = Aut(U) = GL(3) acts on M by left multiplication. It has a subgroup T = (\CC^*)^3 of diagonal matrices and one may restrict the action to this smaller subgroup.

Let M_{na} be the subset of matrices of maximal rank and M_{ab} be the subset of matrices with non-vanishing rows, M_{na} is an open subset in M_{ab}.

Consider quotient spaces X_{na} = M_{na} / G and X_{ab} = M_{ab} / T. Note that X_{na} = Gr(3,V) = Gr(3,7) and X_{ab} = (\PP(V))^3 = (\PP^6)^3.

Since T is a subgroup of G, there is a natural rational map \pi: X_{ab} -> X_{na}: a triple of points in \PP^6 is sent to their linear span.

Weyl group (symmetric group S_3) acts on X_{ab} = (\PP^6)^3 and hence it acts on the cohomology H(X_{ab}) = H((\PP^6)^3), so cohomology space is decomposed into representations of S_3.

[I’ll omit the part of the story with the partial flag space and non-holomorphic map].

1. It turns out that cohomology H(X_{na}) can be identified with antisymmetric part of H(X_{ab}) as a graded vector space (with grading shifted by 3).
Explicitly, H(X_{ab}) is generated by 3 pullbacks H_1, H_2, H_3 of hyperplane sections on \PP^6; cohomologies of Grassmanian are known to be quotient of symmetric polynomials. Vector space of anti-symmetric polynomials is obtained from vector space of symmetric polynomials via multiplication by anti-symmetric polynomial of the smallest degree Formula does not parse: \Delta  = \prod_{i<j} (s_i-s_j)[/latex]. So cohomology class in Grassmanian corresponding to symmetric function [latex]\sigma[/latex] is sent into [latex]\Delta \cup \sigma(H_1,H_2,H_3)[/latex].  <b>2.</b>Also we can compare vector bundles on [latex]X_{ab} and X_{na} by pulling them back to M, and considering as G-linearized.
It turns out that universal bundle U over Gr(3,7) decomposes into sum of 3 line bundles on X_{ab}:  U  <->  O(1,0,0) \oplus O(0,1,0) \oplus O(0,0,1).
So O_{Gr(3,7)}(1) <-> O(1,1,1) and U^*(1) <->  O(1,1,0) \oplus O(1,0,1) \oplus O(0,1,1).

3. On domain of \pi one may define a relative tangent bundle T_{\pi} (“traceless” part of Hom(U,U)). It turns out this vector bundle can be extended as a split vector bundle to whole X_{ab}: T_{\pi} = O(1,-1,0) \oplus O(-1,1,0) \oplus O(1,0,-1) \oplus O(-1,0,1) \oplus O(0,1,-1) \oplus O(0,-1,1).
Consider “square root” of relative tangent bundle t_{\pi} = O(0,-1,1) \oplus O(-1,0,1) \oplus O(-1,1,0).

4.Recall that Fano threefolds V_{22} = V_{na} are sections of homogeneous vector bundle E = 3 U^*(1) on Gr(3,7). Comparision (2) shows these threefolds has 9-dimensional abelianizations V_{ab} — complete intersections of $9$-dimensional split bundle E_{ab} = (O \oplus O \oplus O) \otimes (O(1,1,0) \oplus O(1,0,1) \oplus O(0,1,1)) on X_{ab}.

5. Abelian/non-abelian correspondence is similar for pairs X_{ab}/X_{na} and V_{ab}/V_{na}.
J-series for Gr(3,7) can be obtained as twisted by relative tangent bundle T_{\pi} I-series for e(t_{\pi}) \cup  I_{X_{ab},T_{\pi}} after the comparision of cohomologies described in (1).
Similarly, J-series for V_{22} can be obtained via mirror map from twisted by T_{\pi} + E_{ab} I-series e(t_{\pi}) \cup I_{X_{ab},T_{\pi}+E} after the “pullbacked” comparision (1).

6. The sign comes from considering closely the Gamma-factor for relative tangent bundle T_{\pi}. Note that fibers of abelian/non-abelian correspondence are in some sense holomorphic symplectic (relative tangent bundle contains both O(D) and O(-D)), so they behave like varieties with trivial canonical class.
Consider the factor \frac{\Gamma(1+D+d) \Gamma(1-D-d)}{\Gamma(1+D)\Gamma(1-D)}. Since \Gamma(1+x) \Gamma(1-x) = \frac{\pi x}{sin (\pi x)} and sin (\pi (x+d)) = (-1)^d sin(\pi x) we have \frac{\Gamma(1+D+d) \Gamma(1-D-d)}{\Gamma(1+D)\Gamma(1-D)} = (-1)^d \frac{d+D}{D}.

The same method can also be applied to complete intersections of homogeneous bundles in orthogonal isotropic and symplectic isotropic Grassmanians, since these Grassmanians themselves are just sections of some homogeneous bundles (wedge or symmetric powers of universal bundle) on ordinary Grassmanians of type A. Also this can be uprgaded to treat different blowups of these varieties. In particular, in the comments to this post I compute J-series for Fano threefolds V_5, #2.14, #2.17, #2.20, #2.21 and #2.22.

Also Bumsig points out that one can express the ab/non-ab twist as a differential operator applied to abelian multi-parameter J-function (basically, just Vandermonde \prod_{i>j} (\frac{d}{d q_i} - \frac{d}{d q_j}). This interpretation is more useful for dealing with Frobenius manifolds.

Gromov-Witten Invariants for Abelian and Nonabelian Quotients by Aaron Bertram, Ionut Ciocan-Fontanine, Bumsig Kim
The Abelian/Nonabelian Correspondence and Frobenius Manifolds by Ionut Ciocan-Fontanine, Bumsig Kim, Claude Sabbah
Quantum cohomology of the Grassmannian and alternate Thom-Sebastiani by Bumsig Kim, Claude Sabbah

Beyond Minkowski ansatz

There are two examples of correct polynomials that don’t fit into ansatz stated below.
I think I have shown these examples to Alessio in April.

More info in the notes of my talk (page 2, polynomials w_1 and w_2): pdf (or follow the link from here).

Both examples are degenerations of projective space P^3 (grdb[547386]).

Period sequence 12
First 10 period coefficients: [1, 0, 0, 0, 24, 0, 0, 0, 2520, 0]
The PF operator has N=3, r=4
This sequence has a smooth toric Fano representative
It arises from the following polytopes [(PALP id, grdb id, smoothness)]:
(0, 547386, ‘smooth’)
The PF operator for this sequence is:
256*t^4*D^3 + 1536*t^4*D^2 + 2816*t^4*D + 1536*t^4 – D^3

So we start from the familiar Laurent polynomial
w = x + y + z + \frac{1}{xyz}

I. Argument against “lattice” decomposition.

a. make monomial transformation
m_1: (x,y,z) \to (xz,yz,\frac{1}{z})

w goes to w_1 = z (x+y) + \frac{1}{z} (1+ \frac{1}{xy})

b. mutate by
f_1: (x,y,z) \to (x,y,\frac{x+y}{z})

w_1 goes to \hat{w_1} = \frac{1}{z} + z (x+y) (1 + \frac{1}{xy})

Since \hat{w_1} is derived from w by transformation from group SCr_3 it is a mirror for projective space P^3 (“weak Landau-Ginzburg model” in Przyalkowski’s notations).

Newton polygon of \hat{w_1} is fan polytope of a Gorenstein toric variety – (grdb[544357]).
This variety is anticanonical cone over smooth quadric P^1 \times P^1, i.e. a cone over section of \nu_2(P^3) (P^3 embedded into P^9 by complete linear system of quadrics),
and hence it is a geometric degeneration of P^3.

Consider the quadrangular face, corresponding to a singular point. Restriction of \hat{w_1} to this face is u = z (x+y) (1 + \frac{1}{xy}) = z ( x + y + \frac{1}{x} + \frac{1}{y} ). It is not friendly to Minkowski ansatz’s condition of lattice Minkowski decomposition (this is exactly the example of Minkowski decomposition that is not a lattice Minkowski decomposition given in the definition of the ansatz).

II. Argument against “admissible triangles” and decomposing polytopes completely.

Example w_2 from the same notes.

This one is degeneraiton to P(1,1,2,4) (grdb[547363]).

By monomial transformation

(x,y,z) \to (x,yx,z)

transform w to

w_2 = z + y (x+1) +\frac{1}{z x y^2}

then by mutation

(x,y,z) \to (x, \frac{y}{1+x}, z)

transform w_2 to

\hat{w_2} = z + y + \frac{(1+x)^2}{z x y^2}

P(1,1,2,4) is embedded as a quadric in P(1,1,1,1,2) by linear system O(2), so it is a degeneration of a general quadric in this space i.e. P^3.
This variety P(1,1,2,4) is also the anticanonical cone over singular quadratic surface P(1,1,2).

Restriction to the face equivalent to this surface is equal to
\frac{(x+2+1/x)}{y} + y, so it is not admissible.

Z. How to tune the ansatz?

Universal fix:
allow change of the lattice after creating some of the good polynomials

less universal:
a. Allow non-lattice Minkowski decompositions
b. Increase the set of admissible figures

Update on June 22:
III. Examples further beyond

By combining technique from examples in this and previous post we can construct some more sophisticated
mirrors for Tom and Jerry. These mirrors should correspond to degenerations of these guys to Gorenstein cones over singular (Gorenstein or not) del Pezzo surfaces of degree 6.
I’ll write only numerical details and maybe will provide some geometry later in the comment.

Start from a honeycomb and
u = x+y+\frac{y}{x}+\frac{1}{x}+\frac{1}{y} +\frac{x}{y}

Using cluster transformations it can be transformed to mirrors constructed from Gorenstein toric degenerations of del Pezzo surface S = S_6.

first to pentagon

u_5 = y + x + \frac{1}{x} + \frac{(1+x)^2}{xy}

then to quadruple

u_4 = \frac{1}{xy} + \frac{2}{x} + \frac{2}{y} + \frac{x}{y} + \frac{y}{x} + y

then to triangle

u_3 = xy + 2x + \frac{x}{y} + \frac{3}{y} + \frac{3}{xy} + \frac{1}{x^2y}

The triangle is fan polytope of Gorenstein weighted projective plane P(1,2,3).

We can mutate it further to get non-Gorenstein weighted projective plane P(1,3,8)

u' = y + 3x + 3\frac{(x+1)^2}{y} +\frac{(x+1)^4}{y^2}

Then we choose G equal to 2 or 3
and take

w = z + \frac{u+G}{z}

This will be weak mirror for Jerry or Tom,
all underlying toric threefolds are Gorenstein.

Last two are P(1,2,3,6) (grdb[547331]) and P(1,3,8,12) (grdb[547474]).

Triangles may be Minkowski decomposable only when they are multiples of smaller triangles, which is not the case in these examples.

Altmann’s results on relations between Minkowski decompositions and deformations does not apply here since we have non-isolated singularity (it is a cone over already singular space).

Unsections/Cones and “Tom vs Jerry” ambiguity

Unsections/Cones and “Tom vs Jerry” ambiguity:
why no single-valued ansatz is possible and Minkowski ambiguity is the thing to expect

[Miles Reid-like notation]
Consider two del Pezzo threefolds of degree 6.
Let Jerry be P^1 \times P^1 \times P^1
and Tom be W = P(T_{P^2}) = X_{1,1} \subset P^2 \times P^2 (hyperplane section of product of two planes in Segre embedding).
It is known that Tom and Jerry are not fibers of a flat family.

Tom has period sequence 6,
Jerry is grdb[520140 and has period sequence 21.

Their half-anticanonical section is S = S_6 (del Pezzo surface of degree 6).
So both Tom and Jerry can be degenerated to the same Gorenstein toric Fano threefold — anticanonical cone over S.

This cone has just one integral point except origin and vertices.
Let u = x + y + xy + \frac{1}{x} + \frac{1}{y} +\frac{1}{xy}
be the normalized Laurent polynomial for the honeycomb (fan polytope of S).

Note that honeycomb has two different Minkowski decompositions — as sum of three intervals and as sum of two triangles.

These decompositions correspond to two different decompositions of (u+G) into the product of Laurent polynomials [for two different values of G (G=2 and G+3)]:

u+2 = (1 + x) (1 + y) (1 + \frac{1}{xy})


u+3 = (1 + x + y) (1 + \frac{1}{x} + \frac{1}{y})

General Laurent polynomial for the cone over S
has the form

w_G = z (u + G) + \frac{1}{z}

The most interesting thing is the following:

if we choose $G=2$ then w is mirror of Jerry,
but if we choose $G=3$ then w is mirror of Tom.

Moreover applying mutation we can transform w to terminal Gorenstein polynomials:


z (u+2) + \frac{1}{z} = z (1+x)(1+y)(1+\frac{1}{xy}) + \frac{1}{z}

becomes z (1+x)(1+y) + \frac{(1+\frac{1}{xy})}{z} = z + zx + zy + zxy + \frac{1}{z} + \frac{1}{xyz}

by applying (x,y,z) \to (x,y,\frac{z}{1+\frac{1}{xy}}).

This corresponds to STD of Tom.
It looks nicer after monomial transformation (x,y,z) \to (x,y,\frac{z}{xy}):
\frac{z}{xy} + \frac{z}{x} + \frac{z}{y} + \frac{xy}{z} + \frac{1}{z}


z (u+3) + \frac{1}{z} = z (1+x+y)(1+\frac{1}{x}+\frac{1}{y}) + \frac{1}{z}

becomes z (1+x+y) + \frac{1 + \frac{1}{x} + \frac{1}{y}}{z} = z + zx + zy + \frac{1}{z} + \frac{1}{zx} + \frac{1}{zy}

by applying (x,y,z) \to (x,y,\frac{z}{1+\frac{1}{x}+\frac{1}{y}})

This is simply Laurent polynomial for the smooth model of Jerry:

x+y+z+\frac{1}{x} + \frac{1}{y} + \frac{1}{z}
after monomial transformation
(x,y,z) \to (xz,yz,z).


So Laurent phenomenon distinguishes degenerations of different varieties to the same singular and does not mix them.