# Aspects of enumerative geometry with quadratic forms

## TL;DR

This paper extends classical enumerative geometry by incorporating quadratic forms and Grothendieck-Witt groups, providing new identities and formulas for counting geometric objects with enriched algebraic information.

## Contribution

It introduces a framework replacing numerical formulas with identities in Grothendieck-Witt groups, connecting enumerative geometry with quadratic form theory.

## Key findings

- Formulas for counting degenerate fibers in a pencil using quadratic forms
- Euler characteristic calculations in the context of Grothendieck-Witt groups
- New identities linking classical enumerative invariants with quadratic form theory

## Abstract

We develop various aspects of classical enumerative geometry, including Euler characteristics and formulas for counting degenerate fibres in a pencil, with the classical numerical formulas being replaced by identitites in the Grothendieck-Witt group of quadratic forms with coefficients in the base-field.