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Quadratic counts of highly tangent lines to hypersurfaces

Stephen McKean, Giosuè Muratore, Wern Juin Gabriel Ong

TL;DR

This paper develops an enriched enumerative framework for counting highly tangent lines to smooth hypersurfaces, producing a quadratic-form Euler number valued in the Grothendieck–Witt ring by fixing an orienting divisor. It provides two complementary descriptions: a Wronskian interpretation, where the local index is the trace of the Wronskian of the gradient along pointed lines, and a fundamental-form interpretation, where the index is governed by the second and third fundamental forms of plane curves, culminating in a formula when a theta characteristic exists. The authors establish explicit parameter-space coordinates via the flag variety and construct relative principal parts bundles, analyze their orientability, and implement explicit local trivializations to enable computation of local indices. They also connect the counts to classical inflection theory on plane curves, giving Euler-number computations in the relatively orientable case and describing the appearance of the inflection locus as the zero set of II II. The work situates these counts within enriched enumerative geometry, enabling sign-sensitive, field-general Euler characteristics beyond the complex or real settings.

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.

Quadratic counts of highly tangent lines to hypersurfaces

TL;DR

This paper develops an enriched enumerative framework for counting highly tangent lines to smooth hypersurfaces, producing a quadratic-form Euler number valued in the Grothendieck–Witt ring by fixing an orienting divisor. It provides two complementary descriptions: a Wronskian interpretation, where the local index is the trace of the Wronskian of the gradient along pointed lines, and a fundamental-form interpretation, where the index is governed by the second and third fundamental forms of plane curves, culminating in a formula when a theta characteristic exists. The authors establish explicit parameter-space coordinates via the flag variety and construct relative principal parts bundles, analyze their orientability, and implement explicit local trivializations to enable computation of local indices. They also connect the counts to classical inflection theory on plane curves, giving Euler-number computations in the relatively orientable case and describing the appearance of the inflection locus as the zero set of II II. The work situates these counts within enriched enumerative geometry, enabling sign-sensitive, field-general Euler characteristics beyond the complex or real settings.

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.
Paper Structure (10 sections, 16 theorems, 69 equations, 1 figure)

This paper contains 10 sections, 16 theorems, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline
  3. The parameter space
  4. Coordinates
  5. The case of lines
  6. The bundle
  7. Orienting divisors
  8. Trivializations
  9. Wronskian interpretation of the local index
  10. Highly tangent lines and the second fundamental form

Key Result

Theorem 1.1

Let $k$ be a field. Let $X=\mathbb{V}(F)$ be a smooth, general hypersurface in $\mathbb{P}^n_k$ of degree $d\geq 2n-1$. Let $\Phi$ be the flag variety of pointed lines in $\mathbb{P}^n_k$, and let $\beta\colon \Phi\to\mathbb{P}^n_k$ send a pointed line to its marked point. Let $L$ be a line such tha

Figures (1)

  • Figure 1: Signed counts of real roots

Theorems & Definitions (45)

  • Theorem 1.1: Wronskian interpretation
  • Theorem 1.2: Fundamental form interpretation
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 35 more