Minimal Models and Abundance for Positive Characteristic Log Surfaces

By Hiromu Tanaka. Nagoya mathematical journal, 2012

103 citations recorded by the source corpus.

Abstract We discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for ℚ-factorial surfaces and for log canonical surfaces. Moreover, in the case where the base field is the algebraic closure of a finite field, we obtain the same results under much weaker assumptions.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/849c4dd4b174c9246047c99c4e9c7be3f26d0f61

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Citations in the corpus

Minimal model program for excellent surfaces (2016, background): “Recently, Fujino and the author established the minimal model program for Q-factorial surfaces over algebraically closed fields [6, 30].”
The augmented base locus of real divisors over arbitrary fields (2013, background): “Note that B +(L) is nothing but the union of the ﬁbres of X → Y. Now assume that k has characteristic p > 5 and dimX ≤ 3. One can show that L is again semi-ample (if dimX = 2, this holds for any p [20]). We sketch The augmented base locus of real divisors over arbitrary ﬁelds 15 the proof. Since ”
On the three dimensional minimal model program in positive characteristic (2013, unknown): “igher resolution, we can assume (S′,B S′) is terminal. We then run the minimal model program for (S′,B S′) over Sand we obtain a relative minimal model µ: S′ → S¯ with a morphism ν: S¯ → S (see [KK], [Tanaka12a]). Note that µ∗E∼S,Q KS¯ + BS¯ is nef where BS¯ = µ∗BS′. Since µ∗Eis an exceptional curve”
Existence of flips and minimal models for 3-folds in char p (2013, methodology): “ models for 3-folds in char p 11 and big. Moreover, suppose that each connected component of E(L) is inside some normal irreducible component S of ⌊B⌋. Then L|S is semi-ample for such components (cf. [28]) hence L|E(L) is semi-ample and this in turn implies that L is semi-ample by Theorem 2.9. 3. Ex”
Behavior of canonical divisors under purely inseparable base changes (2015, background): “[Getal15]). In Theorem 1.7 and Theorem 1.8, we establish a cone theorem for surfaces and threefolds over a non-closed ﬁeld. If k is algebraically closed, then Theorem 1.7 and Theorem 1.8 follows from [Tan14] and [CTX], respectively. If X is a smooth projective geometrically connected varietyover ano”
Purely Log Terminal Threefolds with Non-Normal Centres in Characteristic Two (2016, methodology): “Since H 1 ( X, O X ) = 0 [Tan14, Theorem 5.4 and Remark 5.5], we can apply Lemma 3.2.”