## Rank 2 Fano 3-folds

We compute the quantum period sequences of Fano 3-folds in the Mori-Mukai list.

- [Not very Fano]The blow-up of with centre an elliptic curve which is the intersection of two members of . This is a hypersurface in a toric variety . The divisor diagram for is .

Note that is a scroll over with fibre . There is a morphism , which is the blow up along ; this map sends to . There are two line bundles on : are section of ; are sections of ; is a section of and of . is cut out by a section of : we have .Applying quantum Lefschetz gives:

Regularizing this (that is, pre-multiplying by so as to kill the linear term in , and then replacing by ) gives a period sequence that is not in our list:

Note that there are two birational models of the ambient space here, corresponding to the two chambers in the divisor diagram. But the sum defining the I-function really takes place over the intersection of the Mori cones of the two birational models, because of the factor of in the summand. This factor vanishes outside one of the Kahler cones; similarly vanishes outside the other Kahler cone. Similar things happen in many of the examples below.

- [Not very Fano]The double cover of branched along a divisor of bidegree . This is a hypersurface in a toric variety with divisor diagram . With coordinates , the defining equation of is . Denote by the line bundle with sections and by the line bundle with sections ; then .Quantum Lefschetz gives

Regularizing this gives a period sequence that is not in our list:

- [Not very Fano]the blow up of with centre an elliptic curve which is the intersection of two elements of . This is a hypersurface in a toric variety . The divisor diagram for is .

Note that is a scroll over with fibre . There is a morphism , which is the blow up along ; this map sends to . There are two line bundles on : are section of ; are sections of ; is a section of . is cut out by a section of : we have . Quantum Lefschetz gives

Regularizing this gives a period sequence that is not in our list:

- the blow up of with centre an intersection of two cubics. Thus, is a divisor of bidegree on . We denote by and the pull backs of the tautological bundles on the two factors. We have , and quantum Lefschetz gives:

Regularizing this gives period sequence 49:

- the blow up of with centre a plane cubic.This is a hypersurface in a toric variety . The divisor diagram for is .

Note that is a scroll over with fibre . There is a morphism , which is the blow up along ; this map sends to . There are two line bundles on : are section of ; are sections of . is cut out by a section of : we have .Quantum Lefschetz gives

Regularizing this gives period sequence 34:

. - a divisor of bidegree in . Quantum Lefschetz gives

Regularizing this gives period sequence 11:

.

Note that this is a D3 form: even though the Fano has rank 2, the quantum cohomology D-module splits off a “rank 1” irreducible piece (i.e. a piece of dimension 4, which is the size of the cohomology of a rank-1 Fano 3-fold). - the blow up of a quadric with centre the intersection of two members of . Thus, is the complete intersection of two divisors in , of bidegrees and . We denote by and the pull backs of the tautological bundles on the two factors. We have , and quantum Lefschetz gives:

Regularizing this gives period sequence 51:

- a double cover of with branch locus a member of such that is smooth, where is the exceptional divisor of the blow-up . This is a hypersurface in a toric variety . The divisor diagram for is . Call the co-ordinates . Let be the line bundle with sections and let be the line bundle with sections ; is a section of . The variety is a section of on ; we have .Quantum Lefschetz gives:

Regularizing this gives period sequence 26:

- the blow up of in a curve of degree 7 and genus 5. is cut by the equations:

where the are linear forms and the are quadratics. Let . The relations (szyzgies) between these equations are generated by:

Thus is given by these two equations in , where the first factor has co-ordinates and the second factor has co-ordinates . In other words, is a complete intersection in of type ; we have .

Quantum Lefschetz gives:

Regularizing this gives period sequence 62:

- the blow up of with centre an elliptic curve which is the intersection of two hyperplane sections. This is a complete intersection in a toric variety . The divisor diagram for is .

Note that is a scroll over with fibre . There is a morphism , which is the blow up along ; this map sends to . There are two line bundles on : are section of ; are sections of . is a complete intersection of divisors and in ; we have . Quantum Lefschetz gives

Regularizing this gives period sequence 40:

- the blow up of with centre a line on it. This is a hypersurface in a toric variety . The divisor diagram for is .

Note that is a scroll over with fibre . There is a morphism , which is the blow up along ; this map sends to . There are two line bundles on : are section of ; are sections of . is cut out by a section of in ; we have . Quantum Lefschetz gives

Regularizing this gives period sequence 56:

- the blow up of in a curve of degree 6 and genus 3. is cut by the equations:

where the are linear forms. Let be the minors. The relations (szyzgies) between these equations are generated by:

Thus is given by these three equations in , where the first factor has co-ordinates and the second factor has co-ordinates . In other words, is a complete intersection in of type ; we have .

Quantum Lefschetz gives:

Regularizing this gives period sequence 13:

which is a D3 form (because this is obviously a G-Fano: it is Galkin’s ). - the blow-up of a 3-dimensional quadric in a curve of genus 2 and degree 6. This is a complete intersection in a toric variety. We have where the embedding sends to . We blow up inside . The equations defining are

where are co-ordinates on . The blown-up variety is the complete intersection in cut out by the equations:

where are co-ordinates on . Our Fano is the complete intersection of with a quadric . Thus is a complete intersection of type in ; here is the tautological bundle on and is the tautological bundle on .

We have and Quantum Lefschetz gives:

Regularizing this gives period sequence 52:

- the blow-up…
- the blow-up of with center the intersection of a quadric and a cubic. This is a hypersurface in a scroll . The divisor diagram for is

.

The projection sends to . There are two line bundles on : are section of ; are sections of . is cut out of by a section of ; we have . Quantum Lefschetz gives

Regularizing this gives period sequence 35:

- the blow-up of with center a conic on it. This is a complete intersection in a toric variety. We give full details of the construction, as it is a model for several other examples (being a blow-up of a projective hypersurface with center a complete intersection in the ambient projective space).We begin by constructing the blow-up of with center the conic where is a quadratic polynomial in . To do this, introduce new co-ordinates and impose the relation:

Thus we construct as a hypersurface in the toric variety with divisor diagram . The co-ordinates on here are ; the equation defining is . The map from to sends to . It is easy to check that this is the blow-up of in the conic . Introduce line bundles such that are sections of and are sections of ; note that is cut out of by a section of . The Fano is a complete intersection of 3 divisors , , and on . We have . Quantum Lefschetz gives:

Regularizing this gives period sequence 59:

- the blow-up…
- the double cover of with branch locus a divisor of bidegree . This is a hypersurface in a toric variety with divisor diagram . With coordinates , the defining equation of is . Denote by the line bundle with sections and by the line bundle with sections ; then .Quantum Lefschetz gives

Regularizing this gives period sequence 60:

- the blow-up of with center a line on it. We proceed as in example 16; here is a complete intersection in the toric variety with divisor diagram

The co-ordinates on here are ; the map from to sends to . Introduce line bundles such that are sections of and are sections of . Then is a complete intersection of 2 divisors , on . We have . Quantum Lefschetz gives:

Regularizing this gives period sequence 55:

- the blow-up…
- the blow-up…
- the blow-up…
- the blow-up of a quadric with center an intersection of and . is a complete intersection of type in the toric variety with weight data . We have .Quantum Lefschetz gives:

Regularizing this gives period sequence 29:

- A divisor of bidegree on . Quantum Lefschetz gives:

Regularizing this gives period sequence 66:

- The blow up of with centre an elliptic curve which is the complete intersection of two quadrics. is a divisor of bidegree in . Quantum Lefschetz gives:

Regularizing this gives period sequence 28:

- the blow up…
- the blow up of with center a twisted cubic. The twisted cubic in with co-ordinates is given by the condition

Let . The relations (szyzgies) between these equations are generated by:

Thus is given by these two equations in , where the first factor has co-ordinates and the second factor has co-ordinates . In other words, is a complete intersection in of type .

Quantum Lefschetz gives:

Regularizing this gives period sequence 61:

- the blow-up of with centre a plane cubic. is a hypersurface of type in the toric variety with weight data

.

We have . Quantum Lefschetz gives:

Regularizing this gives period sequence 33:

- the blow-up of a quadric 3-fold with centre a conic on it. is a hypersurface of type in the toric variety with weight data

.

We have . Quantum Lefschetz gives:

Regularizing this gives period sequence 42:

- the blow-up of with center a conic. is a hypersurface of type in the toric variety with weight data

.

We have . Quantum Lefschetz gives:

Regularizing this gives period sequence 70:

- the blow-up of a quadric 3-fold with center a line on it. is a hypersurface of type in the toric variety with weight data

.

We have . Quantum Lefschetz gives:

Regularizing this gives period sequence 48:

- a divisor on of bidegree . Quantum Lefschetz gives:

Regularizing this gives period sequence 6:

Note that this is a D3 form. - the blow-up of with center a line. is a toric variety with weight data:

.

We have and:

Regularizing this gives period sequence 54:

- . We have:

Regularizing this gives period sequence 44:

- , which is the blow-up of at a point. This is a toric variety with weight data:

.

We have and:

Regularizing this gives period sequence 30:

- the scroll over . This is a toric variety with weight data:

.

We have and:

Regularizing this gives period sequence 58:

14. This variety X is a section of half-anticanonical class on ,

where is del Pezzo threefold of degree 5.

Regularized I-series for V is

Nonregularized

For line $I_{P^1} = \sum_n \t^{2n} \frac{1}{n!^2}$

For product fourfold we have

So for threefold section X we have to change t^2 to t and do Laplace transform once (for non-regularized):

Regularizing gives

After normalization this gives period sequence 39:

On 17. This is section of half-anticanonical class on , where $E$ is the null-correlation bundle on .

References are

Wisniewski (1989b)

Ruled Fano 4-folds of index 2

Szurek-Wisniewski (1990b)

Fano bundles over and $Q^3$.

Szurek-Wisniewski (1990c)

Fano bundles of rank 2 on and $Q_3$.