scholarly journals Unbounded negativity on rational surfaces in positive characteristic

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Raymond Cheng ◽  
Remy van Dobben de Bruyn
Keyword(s):  
Projective Plane ◽  
Positive Characteristic ◽  
Rational Curves ◽  
Rational Surfaces ◽  
Birational Model

Abstract We give explicit blowups of the projective plane in positive characteristic that contain smooth rational curves of arbitrarily negative self-intersection, showing that the Bounded Negativity Conjecture fails even for rational surfaces in positive characteristic. As a consequence, we show that any surface in positive characteristic admits a birational model failing the Bounded Negativity Conjecture.

scholarly journals Two algebraic deformations of a K3 surface

1981 ◽  
Vol 82 ◽  
pp. 1-26
Author(s):  
Daniel Comenetz
Keyword(s):  
Hyperelliptic Curve ◽  
Double Cover ◽  
Rational Curves ◽  
K3 Surface ◽  
Quartic Surface ◽  
Quadric Cone ◽  
Birational Model ◽  
Affine Line ◽  
Characteristic 2 ◽  
General Member

Let X be a nonsingular algebraic K3 surface carrying a nonsingular hyperelliptic curve of genus 3 and no rational curves. Our purpose is to study two algebraic deformations of X, viz. one specialization and one generalization. We assume the characteristic ≠ 2. The generalization of X is a nonsingular quartic surface Q in P3 : we wish to show in § 1 that there is an irreducible algebraic family of surfaces over the affine line, in which X is a member and in which Q is a general member. The specialization of X is a surface Y having a birational model which is a ramified double cover of a quadric cone in P3.

scholarly journals Linear Systems of Rational Curves on Rational Surfaces

2012 ◽  
Vol 12 (2) ◽  
pp. 261-268-268 ◽  
Author(s):  
D. Daigle ◽  
A. Melle-Hernández
Keyword(s):  
Linear Systems ◽  
Rational Curves ◽  
Rational Surfaces

A note on the cancellation property of k[X, Y]

2015 ◽  
Vol 14 (09) ◽  
pp. 1540007 ◽  
Author(s):  
S. M. Bhatwadekar ◽  
Neena Gupta
Keyword(s):  
Polynomial Ring ◽  
Positive Characteristic ◽  
Perfect Field ◽  
Cancellation Property ◽  
Rational Surfaces ◽  
Field Of Positive Characteristic

In [On affine-ruled rational surfaces, Math. Ann.255(3) (1981) 287–302], Russell had proved that when k is a perfect field of positive characteristic, the polynomial ring k[X, Y] is cancellative. In this note, we shall show that this cancellation property holds even without the hypothesis that k is perfect.

Rational Surfaces with Exceptional Unodes

1967 ◽  
Vol 19 ◽  
pp. 938-951 ◽  
Author(s):  
Patrick du Val
Keyword(s):  
Algebraic Surface ◽  
Rational Curves ◽  
Rational Surfaces ◽  
Image Position

Many years ago, I defined (8) three types of exceptional unode on an algebraic surface, which I called U*8, U*9, U*10, corresponding, on a non-singular model of the surface, to sets of six, seven, and eight rational curves, each of grade — 2, with the intersection patterns represented by the Coxeter-Dynkin graphs now usually known as E6, E7, E8:where each dot represents a curve, and linked dots intersecting curves. In each case we shall denote the curves in the horizontal sequence by S1, s2, … from left to right, and the extra curve meeting s3 by s*.

On the irreducibility of Hilbert scheme of surfaces of minimal degree

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Fedor Bogomolov ◽  
Viktor Kulikov
Keyword(s):  
Projective Plane ◽  
Hilbert Scheme ◽  
Special Class ◽  
Main Idea ◽  
Rational Curves ◽  
Minimal Degree ◽  
Irreducible Components

AbstractThe article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙm+1 is irreducible except m = 4. In the case m = 4 the Hilbert scheme consists of two irreducible components explicitly described in the article. The main idea of our approach is to use the proof of Chisini conjecture [Kulikov Vik.S., On Chisini’s conjecture II, Izv. Math., 2008, 72(5), 901–913 (in Russian)] for coverings of projective plane branched in a special class of rational curves.

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