Enumerating singular curves on surfaces (original) (raw)
Related papers
Enumeration of curves with two singular points
Bulletin des Sciences Mathématiques, 2014
In this paper we obtain an explicit formula for the number of curves in P 2 , of degree d, passing through (d(d + 3)/2 − (k + 1)) generic points and having one node and one codimension k singularity, where k is at most 6. Our main tool is a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle V over M , counted with a sign, is the Euler class of V evaluated on the fundamental class of M. Contents 1 Introduction 1 2 Overview 3 3 Algorithm 8 4 Review of definitions and notations for one singular point 9 5 Transversality 6 Closure and Euler class contribution 7 Euler class computation A Low degree checks B Standard facts about Chern classes and projectivized bundle
Enumeration of curves with one singular point
Journal of Geometry and Physics, 2016
In this paper we obtain an explicit formula for the number of curves in P 2 , of degree d, passing through (d(d + 3)/2 − k) generic points and having a codimension k singularity, where k is at most 7. In the past, many of these numbers were computed using techniques from algebraic geometry. In this paper we use purely topological methods to count curves. Our main tool is a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle V over M , counted with a sign, is the Euler class of V evaluated on the fundamental class of M .
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.
Counting curves in a linear system with upto eight singular points
arXiv: Algebraic Geometry, 2019
In this paper, we develop a systematic approach to enumerate curves with a certain number of nodes and one further singularity which maybe more degenerate. As a result, we obtain an explicit formula for the number of curves in a sufficiently ample linear system, passing through the right number of generic points, that have delta\deltadelta nodes and one singularity of codimension kkk, for all delta+kleq8\delta+k \leq 8delta+kleq8. In particular, we recover the formulas for curves with upto six nodal points obtained by Vainsencher. Moreover, all the codimension seven numbers we have obtained agree with the formulas obtained by Kazarian. Finally, in codimension eight, we recover the formula of A.Weber, M.Mikosz and P.Pragacz for curves with one singular point and we also recover the formula of Kleiman and Piene for eight nodal curves. All the other codimension eight numbers we have obtained are new.
Counting curves on a general linear system with up to two singular points
arXiv (Cornell University), 2015
In this paper we obtain an explicit formula for the number of curves in a compact complex surface X (passing through the right number of generic points), that has up to one node and one singularity of codimension k, provided the total codimension is at most 7. We use a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle V over M , counted with signs, is the Euler class of V evaluated on the fundamental class of M .
Imposing Singular Points and Many Nodes to Curves on Surfaces
Monatshefte für Mathematik, 2002
Let S be a smooth projective surface. Here we study the conditions imposed to curves of a ®xed very ample linear system by a general union of types of singularities when most of connected components of are ordinary double points. This problem is related to the existence of``good'' families of curves on S with prescribed singularities, most of them being nodes, and to the regularity of their Hilbert scheme.
Ordinary Singularities of Algebraic Curves
Canadian mathematical bulletin, 1981
Let A be the local ring at a singular point p of an algebraic reduced curve. Let M (resp. M l5 ..., M h) be the maximal ideal of A (resp. of Â). In this paper we want to classify ordinary singularities p with reduced tangent cone: Spec(G(A)). We prove that G(A) is reduced if and only if: p is an ordinary singularity, and the vector spaces Hom(M"IM? +1 t k) span the vector space Hom(M n /M n+1 , k). If the points of the projectivized tangent cone Proj(G(A)) are in generic position then p is an ordinary singularity if and only if G(A) is reduced. We give an example which shows that the preceding equivalence is not true in general.
Counting singular curves with tangencies
2019
We obtain a recursive formula for the characteristic number of degree ddd curves in mathbbP2\mathbb{P}^2mathbbP2 with prescribed singularities (of type AkA_kAk) that are tangent to a given line. The formula is in terms of the characteristic number of curves with exactly those singularities. Combined with the results of S.~Basu and R.~Mukherjee (\cite{R.M}, \cite{B.R} and \cite{BM8}), this gives us a complete formula for the characteristic number of curves with delta\deltadelta-nodes and one singularity of type AkA_kAk, tangent to a given line, provided delta+kleq8\delta+k \leq 8delta+kleq8. We use a topological method, namely the method of ``dynamic intersections'' (cf.~Chapter 11 in \cite{F}) to compute the degenerate contribution to the Euler class. Till codimension eight, we verify that our numbers are logically consistent with those computed earlier by Caporaso-Harris(\cite{CH}). We also make a non trivial low degree check to verify our formula for the number of cuspidal cubics tangent to a given line, using a resul...
Springer Monographs in Mathematics, 2018
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