## Higher rank Fano 3-folds

# Rank 4 Fanos

- This is divisor of degree (1,1,1,1) on

Regularizing gives period sequence 3:

- the blow-up of the cone over a smooth quadric surface in with center the disjoint union of the vertex and an elliptic curve on . The blow-up of the cone over with center the vertex is the toric variety with weight data:

The morphism to is given by ; the image here is in . To obtain , we blow up the elliptic curve . Thus is the hypersurface in the toric variety with weight data:

Quantum Lefschetz gives:

and regularizing gives period sequence 32:

- The blow-up of with center a curve of tridegree (1,1,2). We can take to be parametrized as . We embed into via the map . Now, in , becomes the complete intersection defined by equations:

Note that the second equation here is just the equation of inside . Thus we can blow up along the locus:

and then construct by imposing the strict transform of the remaining equation . This exhibits as a complete intersection of type inside the toric variety with weight data:

Note that this is rank 4 even though the ambient space has rank 3; there is no contradiction here since the line bundle is not ample and so Lefschetz fails. Quantum Lefschetz (which does not fail) gives:

and regularizing gives period sequence 122:

A cleaner and more systematic development is as follows. The curve is defined scheme-theoretically by the equations:

inside . (Please look at the parametrization given above.) So is given by the equation inside the toric variety with weight data:

Now Quantum Lefschetz gives:

and regularizing gives period sequence 122:

- the blow-up of (rank 3, number 19; the blow-up of a quadric with center two non-colinear points ) with center the strict transform of a conic containing and . Consider the line inside , and also the plane . Let be the blow-up of the line followed by the strict transform of the plane (in that order); this is the toric variety with weight data:

The variety is the strict transform of a general quadric in ; in other words it is a hypersurface of type in . (Note that is rank 4 even though the ambient space is rank 3; there is no contradiction here because is not ample.) Quantum Lefschetz gives:

and regularizing gives period sequence 103:

- The blow-up of with center two disjoint curves, one of bidegree (2,1) and the other of bidegree (1,0). This is a hypersurface of type in the toric variety with weight data:

Quantum Lefschetz gives:

and regularizing gives period sequence 147:

- the blow-up of with center the curve of tridegree (1,1,1). This curve is cut out of by the equations

Thus the blow-up is cut out of the toric variety with weight data:

by the equation:

Quantum Lefschetz gives:

and regularizing gives period sequence 65:

- The blow-up of (a divisor of type (1,1)) with center two disjoint curves on it, of bidegree (0,1) and (1,0). We define as the zero locus of inside . Blowing up the disjoint union of and in induces the blow-up that we seek. Thus is a hypersurface of type in the toric variety with weight data:

Quantum Lefschetz gives:

and regularizing gives period sequence 69:

- the blow-up of with center a curve of tridegree (0,1,1). The curve is cut out of by the equations

and so is cut out of the toric variety with weight data:

Quantum Lefschetz gives:

and regularizing gives period sequence 105:

- This is the toric variety with weight data:

Its regularized period sequence is period sequence 102:

- This is the toric variety with weight data:

Its regularized period sequence is period sequence 142:

- This is the toric variety with weight data:

Its regularized period sequence is period sequence 93:

- This is the toric variety with weight data:

Its regularized period sequence is period sequence 150:

- (degree 26, see the erratum)

This is the blow-up of along a curve of tridegree (1,1,3). Note that is a complete intersection of type , and thus we can realize as a hypersurface of type in the toric variety with weight data:

We have . This is ample on but only semi-positive on the ambient space . Thus we are in the situation described here, and we use the same notation. We have:

Inverting the mirror map gives:

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

We construct the regularized period sequence from this by making the change of variables , , , , :

and then doing the trick with factorials:

This is period sequence 88.

# Rank 5 Fanos

- the blowup of , where is number 29 on the rank-2 list (i.e. the blow-up of the quadric 3-fold with center a conic), with center 3 exceptional lines of the blow-up . To compute this, take the defining equation of the quadric to be

in and blow up the ambient space in the plane . Note that the conic in $\Pi$ contains the co-ordinate points . Blowing up the exceptional lines over these points exhibits as a hypersurface of type in the toric variety with weight data:

The regularized period sequence is:

This is period sequence 114. - This is the toric variety with weight data:

The regularized period sequence is:

This is period sequence 87.

The I-function is the product:

and the regularized period sequence is:

This is period sequence 43.

# Rank > 5 Fanos

These are products of line with del Pezzo surface of degree . The I-series are products of I-series for line and for del Pezzo surfaces.

- . We have:

The regularized period is:

This is period sequence 64. - . We have:

The regularized period is:

This is period sequence 71. - . We have:

The regularized period is:

This is period sequence 45. - [Not very Fano] . We have:

The regularized period is:

This period sequence is non-Gorenstein. - [Not very Fano] . We have:

The regularized period is:

This period sequence is non-Gorenstein.

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