Jose Gonzalez Mathematics Research

Some non-finitely generated Cox rings

This page accompanies the article Some non-finitely generated Cox rings by Jose Gonzalez and Kalle Karu, to appear in Compositio Mathematica.

Below are two lists of projective planes P(a,b,c) whose blowups at the unit element t₀ in the big torus does not have a finitely generated Cox ring:

6814 examples with a,b,c at most N=100. (400KB file)
80133 examples with a,b,c at most N=200. (5MB file)

Here is the C program that produced these lists. (Change N at the beginning, compile with "gcc -lm pplanes1.c" if using Linux.)

Abstract: We give a large family of weighted projective planes, blown up at a smooth point, that do not have finitely generated Cox rings. We then use the method of Castravet and Tevelev to prove that the moduli space M_0,n of stable n-pointed genus zero curves does not have a finitely generated Cox ring if n is at least 13.