Infinitely generated symbolic Rees rings of positive characteristic

TL;DR

This paper investigates the finite generation of Cox rings for certain blow-ups of toric varieties in positive characteristic, providing criteria linked to symbolic Rees rings and specific ideal generators.

Contribution

It establishes new criteria for the finite generation of Cox rings in positive characteristic, connecting geometric properties with algebraic structures like symbolic Rees rings and monomial curve ideals.

Findings

Cox(Y) equals the extended symbolic Rees ring R's(I).

R's(I) is Noetherian iff the characteristic of K is 2 or 3.

The ideal I is generated by 2-minors of a specific 2x3 matrix.

Abstract

Let X be a toric variety over a field K determined by a triangle. Let Y be the blow-up at (1,1) in X. In this paper we give some criteria for finite generation of the Cox ring of Y in the case where Y has a curve C such that C^2 \le 0 and C.E=1 (E is the exceptional divisor). The natural surjection Z^3 \rightarrow Cl(X) gives the ring homomorphism K[Z^3] \rightarrow K[Cl(X)]. We denote by I the kernel of the composite map K[x,y,z] \subset K[Z^3] \rightarrow K[Cl(X)]. Then Cox(Y) coincides with the extended symbolic Rees ring R's(I). In the case where Cl(X) is torsion-free, this ideal I is the defining ideal of a space monomial curve. Let Delta be the triangle (4.1) below. Then I is the ideal of K[x,y,z] generated by 2-minors of the 2*3-matrix {{x^7, y^2, z},{y^{11}, z, x^{10}}}. (In this case, there exists a curve C with C^2=0 and C.E=1. This ideal I is not a prime ideal.) Applying our criteria, we prove that R's(I) is Noetherian if and only if the characteristic of K is 2 or 3.