Enumeration of $n$-fold tangent hyperplanes to a surface

Abstract

For each $1\le n\le 6$ we present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface. This yields in particular the numbers of rational curves in the system of hyperplane sections of a generic $K3-$surface imbedded in \p{n} by a complete system of curves of genus $n$ as well as the number {\bf17,601,000} of rational ({\em singular}) plane quintic curves in a generic quintic threefold.

34 pages, Latex (Corrects Latex errors of previous version, minor changes)

References in corpus (1)

• Projective Degenerations of K3 Surfaces, Gaussian Maps, and Fano Threefolds

Cited by in corpus (38)

• A conjectural generating function for numbers of curves on surfaces
• Rational curves on hypersurfaces [after A. Givental]
• A Proof of the Göttsche-Yau-Zaslow Formula
• Universal polynomials for singular curves on surfaces
• Local Mirror Symmetry: Calculations and Interpretations
• Counting curves of any genus on rational ruled surfaces
• Severi Degrees in Cogenus 4
• M-Theory, Topological Strings and Spinning Black Holes
• Rational curves on Calabi-Yau threefolds
• Enumeration of curves via floor diagrams
• Quantum intersection rings
• Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds
• Geometric Aspects of Mirror Symmetry (with SYZ for Rigid CY manifolds)
• On the enumeration of complex plane curves with two singular points
• Enumeration of curves with one singular point
• Node polynomials for families: results and examples
• Severi degrees on toric surfaces
• Severi Degrees in Cogenus 3
• Enumeration of rational curves in a moving family of ${\mathbb{P}}^2$
• Enumeration of singular varieties with tangency conditions
• Counting planar curves in ${\mathbb{P}}^3$ with degenerate singularities
• Relative Node Polynomials for Plane Curves
• Enumeration of singular hypersurfaces on arbitrary complex manifolds
• Labeled floor diagrams for plane curves
• Computing Severi Degrees with Long-edge Graphs
• Family Gromov-Witten Invariants for Kahler Surfaces
• Genus g Gromov-Witten invariants of Del Pezzo surfaces: Counting plane curves with fixed multiple points
• On the intersections of rational curves with cubic plane curves
• Gopakumar-Vafa BPS invariants, Hilbert schemes and quasimodular forms. I
• Quantum Cohomology at Higher Genus: Topological Recursion Relations and Virasoro Conditions
• Curve Counting à la Göttsche
• Gromov-Witten Theory and Threshold Corrections
• Counting singular curves with tangencies
• Fields of definition of curves of a given degree
• Node Polynomials for Curves on Surfaces
• Counting curves in a linear system with upto eight singular points
• The Aspinwall-Morrison calculation and Gromov-Witten theory
• Counting curves on a general linear system with up to two singular points