Infinitely generated symbolic Rees rings of positive characteristic

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.