The Gonality of Singular Plane Curves (original) (raw)

On the number of singular points of plane curves

This is an extended, renovated and updated report on our joint work [OZ]. The main result is an inequality for the numerical type of singularities of a plane curve, which involves the degree of the curve, the multiplicities and the Milnor numbers of its singular points. It is a corollary of the logarithmic Bogomolov-Miyaoka-Yau's type inequality due to Miyaoka. It was first proven by F. Sakai at 1990 and rediscovered by the authors independently in the particular case of an irreducible cuspidal curve at 1992. Our proof is based on the localization, the local Zariski-Fujita decomposition and uses a graph discriminant calculus. The key point is a local analog of the BMY-inequality for a plane curve germ. As a corollary, a boundedness criterium for a family of plane curves has been obtained. Another application of our methods is the following fact: a rigid rational cuspidal plane curve cannot have more than 9 cusps.

New Asymptotics in the Geometry of Equisingular Families of Curves

International Mathematics Research Notices

Introduction Let D be a divisor on the smooth projective surface Sigma and denote by V = V jDj (S 1 ; : : : ; S r ) the variety of irreducible curves C 2 jDj having exactly r singularities of (topological or analytic) types S 1 ; : : : ; S r . We say that V has the T-property at C 2 V , if the conditions imposed by the individual singularities of C are independent (or transversal), that is, if V is smooth of the expected codimension. For Sigma = P 2 it is well-known that the T-property holds for families of nodal curves (cf. [S]). But for more complicated singularities (beginning with cusps) there are examples, where the T-property fails ([Wa1, Lu, L

A note on a question of Dimca and Greuel

Comptes Rendus Mathematique, 2019

In this note we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and Tjurina numbers of an isolated plane curve singularity in the cases of one Puiseux pair and semi-quasi-homogeneous singularities.

On the Degree of Curves with Prescribed Multiplicities and Bounded Negativity

International Mathematics Research Notices, 2022

We provide a lower bound on the degree of curves of the projective plane mathbbP2\mathbb {P}^2mathbbP2 passing through the centers of a divisorial valuation nu\nu nu of mathbbP2\mathbb {P}^2mathbbP2 with prescribed multiplicities, and an upper bound for the Seshadri-type constant of nu\nu nu, hatmu(nu)\hat {\mu }(\nu )hatmu(nu), constant that is crucial in the Nagata-type valuative conjecture. We also give some results related to the bounded negativity conjecture concerning those rational surfaces having the projective plane as a relatively minimal model.

Points of Low Degree on Smooth Plane Curves

1992

The purpose of this note is to provide some applications of Faltings' recent proof of S. Lang's conjecture to smooth plane curves. Let CCC be a smooth plane curve defined by an equation of degree ddd with integral coefficients. We show that for dge7d\ge 7dge7, the curve CCC has only finitely many points whose field of definition has degree led−2\le d-2led−2 over QQQ, and that for dge8d\ge 8dge8, all but finitely many points of CCC whose field of definition has degree led−1\le d-1led−1 over QQQ arise as points of intersection of rational lines through rational points of CCC.

Slope estimates of Artin-Schreier curves

2001

. For every prime number p coprime to d and f (x) ∈ (Zp ∩ Q)[x], let X/Fp be the Artin-Schreier curve defined by the affine equation y p − y = f (x) mod p. Let NP 1 (X/Fp) be the first slope of the Newton polygon of X/Fp. We prove that there is a Zariski dense subset U in the space A d of degree-d monic polynomials over Q such that for all f (x) ∈ U the following limit exists and limp→∞ NP 1 (X/Fp) = 1 d . This is a "first slope version" of a conjecture of Wan.

Plane curves of minimal degree with prescribed singularities

Inventiones Mathematicae, 1998

We prove that there exists a positive α such that for any integer d ≥ 3 and any topological types S 1 , . . . , Sn of plane curve singularities, satisfying µ(S 1 ) + · · · + µ(Sn) ≤ αd 2 , there exists a reduced irreducible plane curve of degree d with exactly n singular points of types S 1 , . . . , Sn, respectively. This estimate is optimal with respect to the exponent of d. In particular, we prove that for any topological type S there exists an irreducible polynomial of degree d ≤ 14 µ(S) having a singular point of type S.