Quadratic counts of highly tangent lines to hypersurfaces

2025/04/30 by McKean, Stephen, Muratore, Giosuè, Ong, Wern Juin Gabriel
#14G27 #14N10 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.21664

Abstract

We give two geometric interpretations for the local type of a line that is highly tangent to a hypersurface in a single point. One interpretation is phrased in terms of the Wronski map, while the other interpretation relates to the fundamental forms of the hypersurface. These local types are the local contributions of an quadratic form-valued Euler number that depends on a choice of orientation.

Citations

• Hyperflex loci of hypersurfaces
• On fundamental forms and osculating bundles
• Relative Gromov-Witten and maximal contact conics
• On vanishing of fundamental forms of algebraic varieties
• Arithmetic inflection of superelliptic curves
• A¹-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections
• An enriched count of the bitangents to a smooth plane quartic curve
• Real algebraic curves

Related

• Lines highly tangent to a hypersurface
• Unirationality of hypersurfaces via highly tangent lines
• Lines on hypersurfaces
• Hyperflex loci of hypersurfaces
• Highly incidental patterns on a quadratic hypersurface in ℝ⁴