# Algebraic Overshear Density Property

## TL;DR

This paper introduces the algebraic overshear density property, a new concept linking algebraic and holomorphic density properties, and explores its implications for embedding bordered Riemann surfaces into affine surfaces.

## Contribution

It defines the algebraic overshear density property, investigates its consequences, and demonstrates its application in embedding bordered Riemann surfaces.

## Key findings

- The algebraic overshear density property implies both algebraic flexibility and holomorphic density.
- Any bordered Riemann surface embedded in an affine surface with this property admits a proper holomorphic embedding.
- The paper proposes new research directions at the intersection of affine algebraic and elliptic holomorphic geometry.

## Abstract

We introduce the notion of the algebraic overshear density property which implies both the algebraic notion of flexibility and the holomorphic notion of the density property. We investigate basic consequences of this stronger property, and propose further research directions in this borderland between affine algebraic geometry and elliptic holomorphic geometry. As an application, we show that any smoothly bordered Riemann surface with finitely many boundary components that is embedded in a complex affine surface with the algebraic overshear density property admits a proper holomorphic embedding.