A note on a smooth projective surface with Picard number 2

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

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Towards the weighted Bounded Negativity Conjecture for blow-ups of algebraic surfaces

manuscripta mathematica ◽

10.1007/s00229-019-01157-2 ◽

2019 ◽

Vol 163 (3-4) ◽

pp. 361-373

Author(s):

Roberto Laface ◽

Piotr Pokora

Keyword(s):

Line Bundle ◽

Characteristic Zero ◽

Algebraic Surface ◽

Intersection Number ◽

Algebraic Surfaces ◽

The Self ◽

Projective Surface ◽

Intersection Numbers ◽

Weighted Version ◽

Smooth Projective Surface

AbstractIn the present paper we focus on a weighted version of the Bounded Negativity Conjecture, which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a function depending on the intesection of curve with an arbitrary big and nef line bundle that is positive on the curve. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.

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The sheaf of relative differentials of a fibred surface

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100071759 ◽

1993 ◽

Vol 114 (3) ◽

pp. 461-470

Author(s):

Fernando Serrano

Keyword(s):

Linear Systems ◽

Smooth Curve ◽

Base Point ◽

Projective Surface ◽

Positive Part ◽

Rational Singularities ◽

Smooth Projective Surface ◽

Genus 2

AbstractLet Φ: S → C denote a fibration from a smooth projective surface onto a smooth curve, with fibres of genus ≥2. The double dual of the sheaf of relative differentials has been studied by F. Serrano [14]. There, it was proved that dim grows asymptotically as the square of n in case Φ is not isotrivial (i.e. fibres vary in modulus), and the converse holds true in most cases, in a way that can be made precise. In the non-isotrivial case, the present paper provides further information about by analysing the linear systems for large n. If P denotes the positive part of in its Zariski decomposition, then it is shown that |rP| is eventually base-point free for some r > 0. Furthermore, Proj is a normal projective surface, fibred over C, birational to S, and with only rational singularities.

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On Abelian automorphism groups of fibred surfaces of small genus

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100004758 ◽

2001 ◽

Vol 130 (1) ◽

pp. 161-174 ◽

Cited By ~ 3

Author(s):

JIN-XING CAI

Keyword(s):

Automorphism Group ◽

General Type ◽

Automorphism Groups ◽

Projective Surface ◽

Canonical Divisor ◽

Smooth Projective Surface ◽

Small Genus

It is proved that, for a complex minimal smooth projective surface S of general type with a pencil of genus g = 3 or 4, any Abelian automorphism group of S is of order [les ] 12K2S + 96(g − 1), provided K2S > 8(g − 1)2, where KS is the canonical divisor of S.

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On the K-Theoretic Hall Algebra of a Surface

International Mathematics Research Notices ◽

10.1093/imrn/rnaa123 ◽

2020 ◽

Author(s):

Yu Zhao

Keyword(s):

Projective Surface ◽

Hall Algebra ◽

Shuffle Algebra ◽

Coherent Sheaves ◽

Smooth Projective Surface

Abstract In this paper, we define the $K$-theoretic Hall algebra for dimension $0$ coherent sheaves on a smooth projective surface, prove that the algebra is associative, and construct a homomorphism to a shuffle algebra introduced by Negut [ 10].

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Fundamental group schemes of Hilbert scheme of n points on a smooth projective surface

Bulletin des Sciences Mathématiques ◽

10.1016/j.bulsci.2020.102898 ◽

2020 ◽

Vol 164 ◽

pp. 102898

Author(s):

Arjun Paul ◽

Ronnie Sebastian

Keyword(s):

Fundamental Group ◽

Hilbert Scheme ◽

Projective Surface ◽

Group Schemes ◽

Smooth Projective Surface

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THE BOGOMOLOV–MIYAOKA–YAU INEQUALITY FOR LOG CANONICAL SURFACES

Journal of the London Mathematical Society ◽

10.1112/s0024610701002320 ◽

2001 ◽

Vol 64 (2) ◽

pp. 327-343 ◽

Cited By ~ 3

Author(s):

ADRIAN LANGER

Keyword(s):

Vector Bundles ◽

Universal Cover ◽

Kodaira Dimension ◽

Projective Surface ◽

Chern Classes ◽

Singular Surfaces ◽

Smooth Projective Surface ◽

Log Canonical ◽

Surface Singularities ◽

Negative Kodaira Dimension

Let X be a smooth projective surface of non-negative Kodaira dimension. Bogomolov [1, Theorem 5] proved that c21 [les ] 4c2. This was improved to c21 [les ] 3c2 by Miyaoka [12, Theorem 4] and Yau [19, Theorem 4]. Equality c21 [les ] 3c2 is attained, for example, if the universal cover of X is a ball (if κ(X) = 2 then this is the only possibility). Further generalizations of inequalities for Chern classes for some singular surfaces with (fractional) boundary were obtained by Sakai [16, Theorem 7.6], Miyaoka [13, Theorem 1.1], Kobayashi [6, Theorem 2; 7, Theorem 12], Wahl [18, Main Theorem] and Megyesi [10, Theorem 10.14; 11, Theorem 0.1].In [8] we introduced Chern classes of reflexive sheaves, using Wahl's local Chern classes of vector bundles on resolutions of surface singularities. Here we apply them to obtain the following generalization of the Bogomolov–Miyaoka–Yau inequality.

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