The FFRT property of two-dimensional normal graded rings and orbifold curves

Abstract

We study the finite F -representation type (abbr. FFRT) property of a two-dimensional normal graded ring R in characteristic p > 0, using notions from the theory of algebraic stacks. Given a graded ring R, we consider an orbifold curve C, which is a root stack over the smooth curve C = Proj R, such that R is the section ring associated with a line bundle L on C. The FFRT property of R is then rephrased with respect to the Frobenius push-forwards F e * (L i ) on the orbifold curve C. As a result, we see that if the singularity of R is not log terminal, then R has FFRT only in exceptional cases where the characteristic p divides a weight of C.NOBUO HARA AND RYO OHKAWA projective line when C ∼ = P 1 ), which is a sort of Deligne-Mumford stack C with coarse moduli map π : C → C. This is the "minimal covering" of C = Proj R on which D becomes integral. Thus it serves as a very useful tool to deal with rational coefficients divisors (cf. [MO]). The FFRT property of R is then rephrased in terms of a global analogue of FFRT property for a pair (C, L) associated with the line bundle L = O C (π * D). In particular, by studying the structure of the Frobenius push-forwards F e * O C on the orbifold curve C, we prove our results. Let us give an overview of the proof of our main theorem in a bit more detail. The assumption that the singularity of R is not log terminal is equivalent to the condition δ C ≥ 0, where δ C is the degree of the canonical bundle on C. When δ C = 0, we have an étale covering ϕ : E → C from an elliptic curve E, via which the Frobenius push-forward F e * O C is related to that on E. Since the structure of F e* O E on an elliptic curve E is well-understood [A], [Od], we can deduce the result for F e * O C , whose structure differs according to whether E is ordinary or supersingular. When δ C > 0, we prove the stability of F e * O C by the method of [KS], [Su] for non-orbifold curves of genus > 1, from which it follows that F e * O C is indecomposable. This paper is organized as follows. In Section 2 we review some fundamental facts on normal graded rings and root stacks. In Section 3 we rephrase the FFRT property of R = R(C, D) in terms of a global FFRT property on the orbifold curve C constructed from (C, D). Sections 4 and 5 are devoted to the study of weighted projective lines with δ C ≤ 0. In Section 4, we apply Crawley-Boevey's result [CB] to deduce the FFRT property of R = R(P 1 , D) when δ C < 0. In Section 5, we study the case δ C = 0, using a covering ϕ mentioned above, to prove that C does not have global FFRT property. In Section 6, we slightly generalize Sun's result [Su] on the stability of Frobenius push-forwards to orbifold curves with δ C > 0. In Section 7, we summarize the result obtained in the previous sections with the main theorem (Theorem 7.2) and discuss the exceptional cases where p divides denominators of the Q-divisor D.