Things are not as straightforward as they seem

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 FThus 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})
\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.


  1. Sergey says:

    z^{-K_X \cdot d} should be z^{K_X \cdot d}

    Tom, could you please explain if the equivariant redundancy is necessary for the proper mirror map or can be ommited?

    In particular, could you please write what is q_i in the definition of Givental’s I-function.
    There is a redundancy of these parameters: divisors D_1,\dots,D_r form a base for space of T-equivariant divisor classes,
    but are linearly dependant in the non-equivariant Picard group: Formula does not parse: Pic(F) = Pic^T(F)/Hom(T,\C^*).

    Is d any class or a class of effective curve on F and d_i = D_i \cdot d?

    Also, variety F has an action of torus T, but complete intersection X already doesn’t admit such action.
    So -K_X = \sum k_i D_i as an equation in $Pic(X)$ doesn’t define values of k_i,
    should we abandon equivariant parameters in any way and restrict ourselves to the non-equivariant only (in the example this is done indeed)?

  2. Tom says:

    @Sergey: p_1 is the class Poincare-dual to \{x_0=0\}, p_2 is the class Poincare-dual to \{y_0=0\}, and p_3 is the class Poincare-dual to \{s=0\}. (Thus they correspond to the standard basis when I look at the weight data defining the toric variety F.) q_i is dual to p_i. d is an effective class and d_i = p_i \cdot d. As I see it, we do not need to consider the equivariant theory at all.

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