## del Pezzo surfaces

Here are the G-functions ()

and regularized quantum periods ()

for del Pezzo surfaces.

### The del Pezzo surface of degree 9

This is the toric variety . We have:

and the regularized quantum period is:

### The del Pezzo surface of degree 8, case a

This is the toric variety . We have:

and the regularized quantum period is:

### The del Pezzo surface of degree 8, case b

This is the toric variety . We have:

and the regularized quantum period is:

### The del Pezzo surface of degree 7

This is the blow-up of in two points. It is a toric variety. We have:

and the regularized quantum period is:

### The del Pezzo surface of degree 6

This is the blow-up of in three points. It is a toric variety. We have:

and the regularized quantum period is:

### The del Pezzo surface of degree 5

This is a hypersurface of bidegree in . We have:

and the regularized quantum period is:

### The del Pezzo surface of degree 4

This is a (2,2) complete intersection in . We have:

and the regularized quantum period is:

### The del Pezzo surface of degree 3

This is the cubic surface in . We have:

and the regularized quantum period is:

### The del Pezzo surface of degree 2

This is the quartic surface in . We have:

and the regularized quantum period is:

### The del Pezzo surface of degree 1

This is the sextic surface in . We have:

and the regularized quantum period is:

And some mirror images:

## del Pezzo surfaces

Shorter forms for non-normalized -series for del Pezzo surface of degree 6.

From description of as hyperplane section of Segre threefold :

From description of as linear section of Segre fourfold :

Invariant T(X) (defined in these Split notes – Definition 28) for del Pezzo surfaces equals to

Note that potentials I wrote down for del Pezzo surfaces of degrees are _optimal_ – for them .

Potentials for del Pezzo surfaces of degrees 1 and 2 that I wrote down are not optimal: and .

Nevertheless one can obtain from them the optimal potentials by doing a relatively short sequence of mutations (using this script).

Can we beautify this page and stick it together with Fano 3-folds on top?

Done: see

http://coates.ma.ic.ac.uk/fanosearch/?page_id=5657

for the new version, which should also appear in the menu at the top of the blog.