Posts tagged ‘theory’

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

A riddle wrapped in a mystery inside a balls-up

Consider #2 on the Mori-Mukai list of rank-3 Fano 3-folds.  This has been giving us some difficulty, which I have now resolved.  We were making a combination of mistakes.  Mori and Mukai describe the variety X as follows.

A member of |L^{\otimes 2} \otimes_{\cO_{\PP^1 \times \PP^1}}  \cO(2,3)| on the \PP^2-bundle \PP(\cO \oplus  \cO(-1,-1)^{\oplus 2}) over \PP^1 \times \PP^1 such that  X \cap Y is irreducible, where L is the tautological line bundle and Y is a member of |L|.

Our first mistake, as Mukai-sensei pointed out in an email to Corti, was using the wrong weight convention for projective bundles.  Mori and Mukai use negative weights, so the ambient \PP^2-bundle F is the toric variety with weight data:
\begin{array}{cccccccc} x_0 & x_1 & y_0 & y_1 & t_0  & t_1 & t_2   &   \\ 1 & 1 & 0 & 0 & 0  & 1  & 1 & \\ 0& 0 & 1 & 1 & 0  & 1  & 1  & \\ 0 & 0 & 0 & 0& 1  &1 & 1  & \end{array}
For later convenience we change basis, expressing F as the toric variety with weight data:
\begin{array}{cccccccc} x_0 & x_1 & y_0 & y_1 &  t_0  & t_1 & t_2   &   \\ 1 & 1 & 0 & 0 & -1   & 0  & 0 & A \\ 0& 0 & 1 & 1 & -1  & 0   & 0  & B \\ 0 & 0 & 0 & 0& 1  &1 & 1   & C  \end{array}
X is a section of |B+2C|.

Our second mistake was failing to accurately account for the fact that although X is Fano, the bundle A+C (which restricts to -K_X on X) is only semi-positive on F.  Thus we are in the situation described in this post and, in the notation defined there, we have:
\begin{cases} F(q) = 1 \\ G(q) = 2q_3+6q_2 q_3 \\ H_1(q) =  \sum_{b>0} {(-1)^{b} \over b} q_2^b = {-\log(1+q_2)} \\ H_2(q) = {-\log(1+q_2)} \\  H_3(q) = \log(1+q_2) \end{cases}
and hence:
\begin{cases} q_1 = {\hat{q}_1 \over 1 - \hat{q}_2} \\ q_2 = {\hat{q_2} \over 1 - \hat{q_2}} \\ q_3 = \hat{q}_3 (1-\hat{q_2}) \end{cases}
Thus the cohomological-degree-zero part of the J-function is:
\exp({-2}\hat{q}_3(1-\hat{q}_2)-6\hat{q}_2 \hat{q}_3)  \sum_{a,b,c\geq 0} \hat{q}_1^a \hat{q}_2^b  \hat{q}_3^c(1-\hat{q}_2)^{c-b-a} {1 \over z^{a+c}} {(b+2c)! \over a!a!b!b!c!c!(c-b-a)!}
and setting \hat{q}_1 = t, \hat{q}_2 = 1, \hat{q}_3 = t, z=1 yields:
\exp(-6t)  \sum_{a,b\geq 0} t^{2a+b} {(2a+3b)! \over a!a!b!b!(a+b)!(a+b)!}
Regularizing this gives:
I_{reg}(t) =1+58 t^2+600 t^3+13182 t^4+247440 t^5+5212300 t^6+111835920 t^7+ \cdots
As Galkin conjectured, this is period sequence 97.

Expected distribution of equations.

Let X be a smooth Fano threefold with Picard number P = \rho = dim H^2(X).

Then subring of algebraic (even) cycles in X is (2+2P)-dimensional, and its Lefschetz decomposition has P blocks: 1 block of length 4 and (P-1) blocks of length 2. So its image in cohomologies of anticanonical section (K3 surface) is (2+2P – P) = (2+P)-dimensional.

For “general” Fano threefold with Picard number P we expect
regularized quantum differential equation (RQDE) to be of degree (2+P) in D = t \frac{d}{dt}
and to have (2+2P) singular points. Nevertheless degree in t may be more than number of singular points
due to apparent singularities.

It turns out that condition for general is not very general in practice.

Assume Fano threefold X has action of finite group G in one of the 4 ways:
a. G acts on X by regular (algebraic) transformations,
b. G acts on X by symplectic transformations,
c. X is defined over non-algebraically closed field k and G is Galois group Gal(k),
d. X is a fiber of a smooth family over some base B and fundamental group G = \pi_1(B) acts on H^\bullet(X) via monodromy.

For cases a,b,c consider the induced action of G on cohomology of X.

Let p = \rho^G = dim H^2(X)^G be $G$-invariant Picard number.
G-invariant part of cohomologies H(X)^G is (2+2p)-dimensional.

Define \emph{minimal quantum cohomology subring } QH_m(X) of \emph{very small quantum cohomology ring} QH(X) as subring generated by c_1(X) and C[t].
It is easy to see QH_m(X) is contained in H(X, C)^G [t].

This implies that regularized I-series I_X is annihilated by
differential operator of degree (2+p).

So it is natural to ask about possible G-actions on Fano threefolds.
First (numerical) step is to see the possible automorphisms of Mori cone or Kaehler cone.
We have some structures on H^2(X,R):
a. lattice H^2(X,Z) and element c_1(X) inside the lattice,
b. rational polyhedral cone of numerically effective divisors,
c. nondegenerate integral quadratic form (Lefschetz pairing) : (A,B) -> \int_X A \cup B \cup c_1(X).
We call this information \emph{Mori structure}.

Group of automorphisms of Mori structure is finite, and for any action
G-invariant Picard number is not less than dimension of invariants of H^2(X)
with respect to whole group of automorphisms of Mori structure.

As far as I remember (but cannot find a reference) for all Fano threefolds one may find some moduli
and some kind of G-action such that G-invariant Picard group coincides with invariant part of $H^2$ with respect to automorphisms of Mori structure.

The standard reference for automorphisms of Mori structure is probably:
Kenji Matsuki, “Weyl groups and birational transformations among minimal models”, AMS 1995

He studies slightly different problem, but has a similar answer. Unfortunately I haven’t a copy of this book, but copied one page from google books.

He says automorphisms of Mori structures turn out to be Weyl groups.

He claims the following Fano threefolds have nontrivial automorphisms:

P – Picard number, then list of Mori-Mukai numbers with the given Picard number

A_1: 2, 6, 12, 21, 32 (these should be G-Fano, but number 2 is suspicious)
other have p=2

A_2: 1, 27 (G-Fano, suspicious that 13 is in the next line)
A_1: 3, 7, 9, 10, 13, 17, 19, 20, 25, 31 (should correspond to p=2)
other have p=3

A_3: 1 (G-Fano)
A_2: 6 (p=2)
A_1 \times A_1: 2 (p=2)
A_1: 3, 4, 7, 8, 10, 12 (p=3)
trivial – 5,9,11 (should have p=4)
missing number 13 from Erratum

A_1 \times A_2: 3 (p=2)
A_2: 1 (p=3)
A_1: 2 (p=4)

For cases P \geq 6 our threefolds are products of a line and del Pezzo surface P^1 \times S_d. They all have Weyl group of type E_{9-d} and p=2.

So the distribution in p is the following (case 4.13 is missing):
p is always less than 5;
p=4 – 4 varieties: 5.2; 4.5, 4.9, 4.11
p=3 – 26 varieties: 5.1; 4.3, 4.4, 4.7, 4.8, 4.10, 4.12; and 19 with P=3
p=2 – 50 varieties
p=1 – 25 varieties (or 26 if 2.2 is there)

This means just 4 varieties should have N=6, and other have even less.

The obvious thing to do is to recompute ourselves the Mori structure and its automorphisms
(in particular discriminant of Lefschetz quadratic form is an important invariant that we need anyway).

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.

A new ansatz for extremal Laurent polynomials

This post describes a new method for generating Laurent polynomials in 3 variables.  Many of these Laurent polynomials are extremal or of low ramification, and they include the extremal Laurent polynomials mirror to 15 of the 17 minimal Fano 3-folds. We call this method the Minkowksi ansatz.

Let P be a 3-dimensional reflexive polytope.  We will construct a Laurent polynomial with Newton polytope equal to P, or in other words we will explain how to assign a coefficient to each integer point in P.  This goes as follows.

Lattice Minkowski sums

We say that a polygon Q is the lattice Minkowski sum of polygons R and S if and only if both:

  • Q = R + S, so that Q is the Minkowski sum of R and S as usual
  • the integer lattice in Q is the sum of the integer lattices in R and S.

Note that any of the the polygons Q, R, S here are allowed to be degenerate.

Examples:  here are two lattice Minkowksi decompositions P = Q+R of a hexagon:

A lattice Minkowksi decomposition

Another lattice Minkowski decomposition

Note that the same lattice polygon can have more than one lattice Minkowski decomposition.  Note also that the first decomposition here is not a complete decomposition into lattice-Minkowksi-irreducible pieces, because the square Q can be further decomposed as the sum of a vertical and a horizontal line.

This is not a lattice Minkowski decomposition

The example above is not a lattice Minkowski decomposition, because the lattice in P is not the sum of the lattices in Q and R.  In fact P is lattice Minkowski irreducible.

Decompose the facets into irreducible pieces

There are 4319 3-dimensional reflexive polytopes.  These polytopes contain a total of 344 distinct facets, where we regard two facets as the same if and only if they differ by a lattice-preserving automorphism.  Of these facets, 79 are lattice Minkowski irreducible.  These 79 facets are also the non-degenerate polygons which occur when the 344 total facets are decomposed into lattice Minkowksi irreducible pieces.  Of those 79 facets, exactly 8 contain no interior lattice points.  Those 8 triangles, which we call admissible triangles are all of type A_n:

The eight admissible triangles

In other words, the cones over these triangles give affine toric varieties that are transverse A_n singularities, for 1 \leq n \leq 8.

The ansatz

Given a 3-dimensional reflexive polytope P, we construct a possibly-empty list of Laurent polynomials as follows.  For each facet F of P, decompose F into lattice-Minkowksi-irreducible pieces in all possible ways.  Discard any such decomposition of F which contains a non-degenerate polygon that is not an admissible triangle.  Any remaining decomposition of F will consist of line segments and admissible triangles.  To this decomposition we associate a Laurent polynomial which is the product of certain basic Laurent polynomials corresponding to line segments and  to admissible triangles.  The basic Laurent polynomials for admissible triangles are:

The coefficients of the basic Laurent polynomials for admissible triangles.

and so on for the remaining admissible triangles.  The basic Laurent polynomials for line segments are:

The coefficients of the basic Laurent polynomials for line segments

and so on for other line segments.

So now, for each facet F of P, we have a list L_F of Laurent polynomials; this list will be empty if F cannot be written as a lattice Minkowksi sum of line segments and admissible triangles. In other words for each facet F we have list of ways of assigning coefficients to each integer point in F.  We seek a list of Laurent polynomials with Newton polytope equal to P, or in other words a list of ways of assigning coefficients to each integer point in P.  This is produced by assigning the coefficient zero to the origin (which is the only interior point of P) and then assigning coefficients to the integer points on facets of P as specified in the facet lists (but amalgamated in all possible ways, so if there are n_F elements in the list for facet F then the number of elements in the list for P is \prod_{\text{facets F}} n_F).

Points to Note

  • This ansatz almost generalizes the earlier recipes given by Pryjzalkowski and Galkin, but differs a little because of the difference between Minkowski decomposition and lattice Minkowksi decomposition.
  • Altman has studied the deformation theory of affine toric varieties and discovered a close connection with Minkowski decompositions.  Since we expect to find the local system associated to an extremal Laurent polynomial f as a piece of the quantum cohomology local system associated to a smoothing of the Newton polytope of f, this is encouraging.  But note that Minkowski decomposition and lattice Minkowksi decomposition are not the same.
  • We suspect that if P is a 3-dimensional reflexive polytope containing a facet with no admissible lattice Minkowski decompositions then the toric variety corresponding to P does not smooth.  More on this later.
  • This ansatz also fits well with  Kouchnirenko’s criterion for a Laurent polynomial to be degenerate.

(I learned this last point from Hiroshi.)