## Things are not as straightforward as they seem

Consider the blow-up of with center a complete intersection of type . Since the complete intersection consists of three points, is a del Pezzo surface . 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 as a complete intersection in a toric variety as follows. Let have weight data:

Now consider the equation:

where and are polynomials in of bidegrees (respectively) (1,1) and (2,1). The variety defined by this equation is cut out by a section of the line bundle ; by projecting we see that is, as desired, the blow-up of in a complete intersection of type . We have and:

Regularizing gives the period sequence:

**This is not correct**: we know the regularized period sequences for del Pezzo surfaces, and in the case of we get:

So what went wrong?

The construction of given above is correct. It is the second half of the calculation which is flawed. The key point is that, even though is Fano and is cut out of the ambient space by a section of an ample line bundle, is not the restriction of an ample line bundle on but rather is only the restriction of a *semi-positive* line bundle on . *Thus the mirror map is non-trivial.* To see this we need to consider not Golyshev’s I-function:

but rather the full Givental I-function:

Here is cut out of the toric variety with toric divisors by a section of the direct sum of line bundles . Golyshev’s I-function is obtained from Givental’s I-function by taking the term in cohomological degree zero and setting:

where . Note that Givental’s I-function is homogeneous of degree zero if we set , , and whenever .

In the situation at hand (i.e. is a semipositive complete intersection in a toric variety) we have, for grading reasons:

where are degree-zero power series in the and is a degree-1 power series in the . Furthermore Givental’s mirror theorem states that:

where:

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 , , and doing the trick with factorials: .

Applying this discussion in our case ( realized as above) yields:

and hence:

Thus the cohomological-degree-zero part of the J-function is:

and setting , , , yields:

Regularizing this gives:

which agrees with our previous calculation.