# Minimal degree equations for curves and surfaces (variations on a theme of Halphen)

## TL;DR

This paper explores extremal properties of algebraic curves and surfaces related to their minimal degrees and the number of hypersurfaces passing through them, contributing new insights into classical algebraic geometry problems.

## Contribution

It introduces new results on minimal degree equations for curves and surfaces, extending classical theories and posing open questions for further research.

## Key findings

- Characterization of minimal degree equations for curves and surfaces
- Bounds on the number of hypersurfaces passing through varieties
- New insights into extremal behaviors in algebraic geometry

## Abstract

Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such collections of varieties: their minimal degree and their maximal number of linearly independent smallest degree hypersurfaces passing through them. We show results for curves and surfaces, and pose several questions.