scholarly journals Automorphisms of quasi-projective surfaces over fields of finite characteristic

2021 ◽  
Author(s):  
Alexandra Kuznetsova
Keyword(s):  
Finite Characteristic ◽  
Projective Surfaces

scholarly journals Conjectures on Stably Newton Degenerate Singularities

2021 ◽  
Author(s):  
Jan Stevens
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Number ◽  
Milnor Number ◽  
Plane Curves ◽  
Newton Diagram ◽  
Finite Characteristic ◽  
Vanishing Cycles

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

scholarly journals Finite groups of units of finite characteristic rings

2017 ◽  
Vol 197 (3) ◽  
pp. 661-671 ◽  
Author(s):  
Ilaria Del Corso ◽  
Roberto Dvornicich
Keyword(s):  
Finite Groups ◽  
Finite Characteristic

scholarly journals From partitions to Hodge numbers of Hilbert schemes of surfaces

2019 ◽  
Vol 378 (2163) ◽  
pp. 20180435 ◽  
Author(s):  
Nate Gillman ◽  
Xavier Gonzalez ◽  
Ken Ono ◽  
Larry Rolen ◽  
Matthew Schoenbauer
Keyword(s):  
Partition Function ◽  
Royal Society ◽  
Asymptotic Properties ◽  
Circle Method ◽  
100Th Anniversary ◽  
Hilbert Schemes ◽  
Hodge Numbers ◽  
Projective Surfaces

We celebrate the 100th anniversary of Srinivasa Ramanujan's election as a Fellow of the Royal Society, which was largely based on his work with G. H. Hardy on the asymptotic properties of the partition function. After recalling this revolutionary work, marking the birth of the ‘circle method’, we present a contemporary example of its legacy in topology. We deduce the equidistribution of Hodge numbers for Hilbert schemes of suitable smooth projective surfaces. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.

scholarly journals Rational cuspidal curves in projective surfaces. Topological versus algebraic obstructions

2017 ◽  
Vol 28 (14) ◽  
pp. 1750106
Author(s):  
Maciej Borodzik
Keyword(s):  
Floer Homology ◽  
Singular Points ◽  
The Other ◽  
Heegaard Floer Homology ◽  
Heegaard Floer ◽  
Bezout Theorem ◽  
Projective Surfaces

We study rational cuspidal curves in projective surfaces. We specify two criteria obstructing possible configurations of singular points that may occur on such curves. One criterion generalizes the result of Fernandez de Bobadilla, Luengo, Melle–Hernandez and Némethi and is based on the Bézout theorem. The other one is a generalization of the result obtained by Livingston and the author and relies on Ozsváth–Szabó inequalities for [Formula: see text]-invariants in Heegaard Floer homology. We show by means of explicit calculations that the two approaches give very similar obstructions.

scholarly journals Del Pezzo orders on projective surfaces

2003 ◽  
Vol 173 (1) ◽  
pp. 144-177 ◽  
Author(s):  
Daniel Chan ◽  
Rajesh S. Kulkarni
Keyword(s):  
Del Pezzo ◽  
Projective Surfaces
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