<i>F</i>-blowups of normal surface singularities

Hara, Nobuo; Sawada, Tadakazu; Yasuda, Takehiko

doi:10.2140/ant.2013.7.733

Cited by 6 publications

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References 18 publications

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“…We divide the proof into two cases, according to whether E is ordinary or supersingular. First we recall the following: Lemma 5.4 ( [A], [HSY,Lemma 4.12]). Let E be an elliptic curve in characteristic p and let q = p e for e ≥ 0.…”

Section: Proofmentioning

confidence: 99%

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

Hara

Ohkawa

2017

Preprint

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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.

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Section: Proofmentioning

confidence: 99%

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

Hara

Ohkawa

2017

Preprint

Self Cite

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“…The main novelty in Villamayor's construction is that he showed that there exists a module E 1 of projective dimension at most 1 having rank e whose Fitting ideal F e (E 1 ) is a representative of [[E]] R , as a fractional ideal. This Fitting ideal F e (E 1 ) is in fact a sub-determinantal ideal of any matrix representing E, which gives an effective method to compute the blow up of R at E. For instance, N. Hara, T. Sawada and T. Yasuda have recently used this approach in [5] in order to compute explicitly F -blowups for some surface singularities by using Macaulay2 [10].…”

Section: Introductionmentioning

confidence: 99%