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Things are not as straightforward as they seem

December 3, 2010, 2:19 pm

Consider the blow-up X of \PP^1 \times \PP^1 with center a complete intersection of type (2,1)\cdot(1,1).  Since the complete intersection consists of three points, X is a del Pezzo surface dP_5.  It is tempting to compute its regularized period sequence as follows.

Warning: this calculation is wrong. I explain below where the error is and how to fix it.  We express X as a complete intersection in a toric variety F as follows.  Let F have weight data:
\begin{array}{ccccccc} x_0 & x_1 & y_0 & y_1 & s & t & \\ 1 & 1 & 0 & 0 & 0 & -1 & L \\ 0 & 0 & 1 & 1 & 0 & 0 & M \\ 0 & 0 & 0 & 0 & 1 & 1 & N \end{array}
Now consider the equation:
s f_{1,1} + t g_{2,1} = 0
where f_{1,1} and g_{2,1} are polynomials in x_i, y_j of bidegrees (respectively) (1,1) and (2,1).  The variety X defined by this equation is cut out by a section of the line bundle L+M+N; by projecting [x_0:x_1:y_0:y_1:s:t] \mapsto [x_0:x_1:y_0:y_1] we see that X is, as desired, the blow-up of \PP^1 \times \PP^1 in a complete intersection of type (2,1)\cdot(1,1).  We have -K_X = M+N and:
I_X(t) = \sum_{l,m,n \geq 0} t^{m+n} {(l+m+n)! \over l!l!m!m!n!(n-l)!}
Regularizing gives the period sequence:
I_{reg}(t) = 1-3 t+23 t^2-105 t^3+783 t^4-4053 t^5+29729 t^6+\cdots

This is not correct: we know the regularized period sequences for del Pezzo surfaces, and in the case of dP_5 we get:
I_{reg}(t) =1+12 t^2+42 t^3+468 t^4+3360 t^5+31350 t^6 + \cdots
So what went wrong?

The construction of X given above is correct.  It is the second half of the calculation which is flawed.  The key point is that, even though X is Fano and is cut out of the ambient space F by a section of an ample line bundle, -K_X is not the restriction of an ample line bundle on F but rather is only the restriction of a semi-positive line bundle on F.  Thus the mirror map is non-trivial. To see this we need to consider not Golyshev’s I-function:
I_X(q) = \sum_{d} q^{-K_X\cdot d} {\prod_{i} (E_i\cdot d)! \over \prod_j (D_j \cdot d)!}
but rather the full Givental I-function:
I^{Giv}_X(q) = q_1^{D_1/z}\cdots q_r^{D_r/z} \sum_{d} q_1^{d_1}\cdots q_r^{d_r} \prod_{i} {\Gamma(1+E_i/z+E_i\cdot d) \over \Gamma(1+E_i/z)} \prod_j {\Gamma(1+D_j/z) \over \prod_j \Gamma(1+D_j/z+D_j \cdot d)} z^{K_X \cdot d}
Here X is cut out of the toric variety with toric divisors D_j by a section of the direct sum of line bundles \oplus_i E_i.  Golyshev’s I-function is obtained from Givental’s I-function by taking the term in cohomological degree zero and setting:
\begin{cases} q_1 = q^{k_1} \\ \vdots \\ q_r = q^{k_r} \\ z = 1 \end{cases}
where -K_X = k_1 D_1 + \ldots + k_r D_r.  Note that Givental’s I-function is homogeneous of degree zero if we set \deg q_i = k_i, \deg z = 1, and \deg \alpha = m whenever \alpha \in H^{2m}(X).

In the situation at hand (i.e. X is a semipositive complete intersection in a toric variety) we have, for grading reasons:
I^{Giv}_X(q) = q_1^{D_1/z}\cdots q_r^{D_r/z} \Big(F(q) + G(q)/z + H_1(q) D_1/z + \cdots + H_r(q) D_r/z + O(z^{-2}) \Big)
where F, H_1,\ldots,H_r are degree-zero power series in the q_i and G is a degree-1 power series in the q_i.  Furthermore Givental’s mirror theorem states that:
{exp\Big(-{G(q) \over z F(q)}\Big) \over F(q)} I^{Giv}_X(q) = J_X(\hat{q})
where:
\begin{cases} \hat{q}_1 = q_1 \exp(H_1(q)/F(q)) \\ \vdots \\ \hat{q}_r = q_r \exp(H_r(q)/F(q)) \end{cases}
This change of variables is called the mirror map.  The regularized quantum period sequence that we seek is obtained from the cohomological-degree-zero  component of the J-function by setting \hat{q}_i = t^{k_i}, z=1, and doing the trick with factorials: \sum_k a_k t^k \longmapsto \sum_k k! a_k t^k.

Applying this discussion in our case (X = dP_5 realized as above) yields:
\begin{cases} F(q) = 1 \\ G(q) = q_2+q_3+2q_1 q_3 \\ H_1(q) = \sum_{k>0} {(-1)^{k} \over k} q_1^k = {-\log(1+q_1)} \\ H_2(q) = 0 \\ H_3(q) = \log(1+q_1) \end{cases}
and hence:
\begin{cases} q_1 = {\hat{q}_1 \over 1 - \hat{q}_1} \\ q_2 = \hat{q_2} \\ q_3 = \hat{q}_3 (1-\hat{q_1}) \end{cases}
Thus the cohomological-degree-zero part of the J-function is:
\exp(-\hat{q}_2-\hat{q}_3(1-\hat{q}_1)+2\hat{q}_1 \hat{q}_3) \sum_{k,l,m\geq 0} \hat{q}_1^k \hat{q}_2^l \hat{q}_3^m(1-\hat{q}_1)^{m-k} {1 \over z^{l+m}} {(k+l+m)! \over k!k!l!l!m!(m-k)!}
and setting \hat{q}_0 = 1, \hat{q}_1 = t, \hat{q}_2 = t, z=1 yields:
\exp(-3t) \sum_{k,l\geq 0} t^{k+l} {(2k+l)! \over k!k!l!l!k!}
Regularizing this gives:
I_{reg}(t) =1+12 t^2+42 t^3+468 t^4+3360 t^5+31350 t^6 + \cdots
which agrees with our previous calculation.

Category: Uncategorized  |  2 Comments

More pictures

December 3, 2010, 12:25 pm

Here are some more pictures of the anticanonical surfaces inside some of our Fano 3-folds. Once again, thanks to Andrew MacPherson for the images.

The quadric 3-fold
Rank 2 #2
Rank 3, #9
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Pictures

November 22, 2010, 2:10 pm

Here are some pictures of the anticanonical surfaces inside some of our Fano 3-folds.  In each case we express the Fano X as a complete intersection in a toric variety Y, choose an affine chart on Y, take the intersection of an anticanonical hypersurface with that affine chart, and then take a generic projection of the resulting surface into affine 3-space.  Thanks to Andrew MacPherson for the images.

the rank-1 Fano 3-fold V_6
The cubic 3-fold
Rank 3 #1
Rank 3, #4
Rank 4, #2
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del Pezzo surfaces

November 16, 2010, 4:19 pm

Here are the G-functions (I = G_{+s} = e^s \cdot G)
and regularized quantum periods (I_{reg} = \hat{G})
for del Pezzo surfaces.

The del Pezzo surface of degree 9

This is the toric variety \PP^2.  We have:
G(t) = \sum_k {t^{3k} \over k!k!k!} = 1+t^3+\frac{t^6}{8}+\frac{t^9}{216}+\cdots
and the regularized quantum period is:
\hat{G}(t) = 1+6 t^3+90 t^6+1680 t^9+\cdots

The del Pezzo surface of degree 8, case a

This is the toric variety \PP^1 \times \PP^1. We have:
G(t) = \sum_{k,l} {t^{2k+2l} \over k!k!l!l!} = 1+2 t^2+\frac{3 t^4}{2}+\frac{5 t^6}{9}+\cdots
and the regularized quantum period is:
\hat{G}(t) = 1+4 t^2+36 t^4+400 t^6+\cdots

The del Pezzo surface of degree 8, case b

This is the toric variety \mathbb{F}_1. We have:
G(t) = \sum_{k,l} {t^{2k+l} \over k!l!l!(k-l)!} = 1+t^2+t^3+\frac{t^4}{4}+\frac{t^5}{2}+\frac{11 t^6}{72}+\frac{t^7}{12}+\cdots
and the regularized quantum period is:
\hat{G}(t) = 1+2 t^2+6 t^3+6 t^4+60 t^5+110 t^6+420 t^7+\cdots

The del Pezzo surface of degree 7

This is the blow-up of \PP^2 in two points. It is a toric variety. We have:
G(t) = \sum_{k,l,m} {t^{2k+3l+2m} \over (l+m)!(k+l)!m!l!k!} =1+2 t^2+t^3+\frac{3 t^4}{2}+t^5+\frac{49 t^6}{72}+\frac{5 t^7}{12}+\cdots
and the regularized quantum period is:
\hat{G}(t) = 1+4 t^2+6 t^3+36 t^4+120 t^5+490 t^6+2100 t^7+\cdots

The del Pezzo surface of degree 6

This is the blow-up of \PP^2 in three points. It is a toric variety. We have:
G(t) = \sum_{k,l,m,n} {t^{k+2l+2m+n} \over (-k+m+n)!(k+l-n)!n!m!l!k!} =1+3 t^2+2 t^3+\frac{15 t^4}{4}+3 t^5+\frac{17 t^6}{6}+2 t^7+\cdots
and the regularized quantum period is:
\hat{G}(t) = 1+6 t^2+12 t^3+90 t^4+360 t^5+2040 t^6+10080 t^7+\cdots

The del Pezzo surface of degree 5

This is a hypersurface of bidegree (1,2) in \PP^1 \times \PP^2. We have:
G_{+3}(t) = \sum_{k,l} t^{k+l} {(k+2l)! \over k!k!l!l!l!} = 1+3 t+\frac{19 t^2}{2}+\frac{49 t^3}{2}+\frac{417 t^4}{8}+\frac{3751 t^5}{40}+\frac{104959 t^6}{720}+\frac{334769 t^7}{1680}+\cdots
and the regularized quantum period is:
\hat{G}(t) = 1+10 t^2+30 t^3+270 t^4+1560 t^5+11350 t^6+77700 t^7+\cdots

The del Pezzo surface of degree 4

This is a (2,2) complete intersection in \PP^4. We have:
G_{+4}(t) = \sum_k t^{k} {(2k)!(2k)! \over k!k!k!k!k!} = 1+4 t+18 t^2+\frac{200 t^3}{3}+\frac{1225 t^4}{6}+\frac{2646 t^5}{5}+\frac{5929 t^6}{5}+\frac{81796 t^7}{35}+\cdots
and the regularized quantum period is:
\hat{G}(t) = 1+20 t^2+96 t^3+1188 t^4+10560 t^5+111440 t^6+1142400 t^7+\cdots

The del Pezzo surface of degree 3

This is the cubic surface in \PP^3. We have:
G_{+6}(t) = \sum_k t^{k} {(3k)!\over k!k!k!k!} = 1+6 t+45 t^2+280 t^3+\frac{5775 t^4}{4}+\frac{63063 t^5}{10}+\frac{119119 t^6}{5}+\frac{554268 t^7}{7}+\cdots
and the regularized quantum period is:
\hat{G}(t) = 1+54 t^2+492 t^3+9882 t^4+158760 t^5+2879640 t^6+51982560 t^7+\cdots

The del Pezzo surface of degree 2

This is the quartic surface in \PP(1,1,1,2). We have:
G_{+12}(t) = \sum_k t^{k} {(4k)!\over k!k!k!(2k)!} = 1+12 t+210 t^2+3080 t^3+\frac{75075 t^4}{2}+\frac{1939938 t^5}{5}+\frac{52055003 t^6}{15}+\frac{191222460 t^7}{7}+\cdots
and the regularized quantum period is:
\hat{G}(t) = 1+276 t^2+6816 t^3+314532 t^4+12853440 t^5+569409360 t^6+25533244800 t^7 +\cdots

The del Pezzo surface of degree 1

This is the sextic surface in \PP(1,1,2,3). We have:
G_{+60}(t) = \sum_k t^{k} {(6k)!\over k!k!(2k)!(3k)!} = 1+60 t+6930 t^2+680680 t^3+\frac{111546435 t^4}{2}+3881815938 t^5+233987238485 t^6+\frac{86928646722060 t^7}{7}+\cdots
and the regularized quantum period is:
\hat{G}(t) = 1 + 10260 t^2 + 2021280 t^3 + 618874020 t^4 + 184450426560 t^5 + 57876331467600 t^6 + 18570232920355200 t^7+\cdots

Category: Uncategorized  |  5 Comments

I’ve fixed an annoying bug in LaTeX output

October 13, 2010, 1:14 pm

I’ve fixed a bug in the wp-LaTeX plug-in which was preventing certain matrices displaying correctly.  All the LaTeX output on the site now appears to render correctly.

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We’ve gone all Web 2.0

October 9, 2010, 7:54 pm

I’ve set up a Twitter account @fanosearch and arranged for blog entries and comments to appear automatically as tweets. Also results of major computer calculations will appear as tweets (or at least they will once I’ve figured out how to use the Twitter API from Sage).

If you have a Twitter account, you can follow @fanosearch to get these updates.  Or you can get them direct from your web browser by subscribing to this RSS feed.  If you prefer to get the blog posts and comments separately then you can use these RSS feeds (posts, comments) which come direct from the blog.  To add the @fanosearch Twitter feed to Facebook, use this.

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V. Batyrev: “Toric Deformations of Some Spherical Fano Varieties”

September 22, 2010, 12:09 pm

Here are my (incomplete) notes from Batyrev’s talk in the Extremal Laurent Polynomials workshop at Imperial:

Batyrev_London_Sep_2010

If you have complete notes, please post them here.

Partial summary:

  • spherical varieties
  • a model family of examples: SL_2-varieties
  • review: compactification and toric varieties
  • spherical compactification in these examples
  • valuations and coloured cones
  • finding toric degenerations
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K. Altmann: “Deformations of Gorenstein Canonical Toric Singularities”

September 22, 2010, 12:04 pm

Here are my notes from Altmann’s talk at the Extremal Laurent Polynomials workshop at Imperial:

Altmann_London_Sep_2010

Summary:

  • an example due to Pinkham
  • deformations and Minkoski sums; a lattice condition
  • constructing the deformation corresponding to a Minkwoski decomposition
  • the versal deformation space and the moduli space of generalized Minkowski summands of Q
  • Example: the cone over a hexagon
  • the equations defining the versal deformation space in the isolated Gorenstein case
  • a new point of view: double divisors
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B. Siebert: “A Tropical View on Landau-Ginzburg Models”

September 22, 2010, 11:58 am

Here are my notes from Siebert’s talk at the Extremal Laurent Polynomials workshop at Imperial:

Siebert_London_Sep_2010

Summary:

  • An overview of toric degenerations and the Gross–Siebert picture
  • Examples: a pencil of quartics in \PP^3 and a pencil of elliptic curves in \PP^2
  • the Gross–Siebert Reconstruction Theorem
  • Mirror Symmetry and the discrete Legendre Transform
  • Landau–Ginzburg models; the Hori–Vafa mirror
  • how to extend the superpotential from the central fiber to the whole of the mirror family in such a way that the resulting superpotential is proper
  • Example: \PP^2, flattening the boundary of the polyhedral complex
  • Broken lines and scattering
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V. Golyshev: “The Apery Class and the Gamma Class”

September 22, 2010, 11:50 am

Here are my notes from Golshev’s talk at the Extremal Laurent Polynomials workshop at Imperial:

Golyshev_London_Sep_2010

Summary:

  • a historical analogy: the study of Fano varieties now versus the study of algebraic varieties in the early 1970s
  • the key idea: classifying Fanos by detecting and classifying Fano quantum motives and their realizations
  • possible approaches
  • the Tannakian picture
  • the quantum Satake correspondence (Golyshev–Manivel)
  • two steps in this direction:  Ueda’s proof of the Dubrovin Conjecture for Gr(k,n); Galkin–Golyshev–Iritani’s proof that Apery=Gamma
  • irregular monodromy data for Dubrovin’s quantum connection
  • the Extended Dubrovin Conjecture and the Apery=Gamma hypothesis
  • the Extended Dubrovin Conjecture and the Apery=Gamma hypothesis hold for Gr(k,n)
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