The Betti table of a high degree curve is asymptotically pure

TL;DR

This paper proves that for high degree curves, the dominant component of their Betti table simplifies to a pure form determined solely by the genus, resolving a question posed by Ein and Lazarsfeld.

Contribution

It establishes that the Boij--S"oderberg decomposition of high degree curves becomes asymptotically pure, depending only on the genus, which is a new understanding in algebraic geometry.

Findings

Main term of the Betti table is a single pure diagram for high degree curves.

The pure diagram depends only on the genus of the curve.

Answers a previously open question by Ein and Lazarsfeld.

Abstract

We prove that asymptotically in the degree, the main term of the Boij--S\"oderberg decomposition of a high degree curve is a single pure diagram that only depends on the genus of the curve. This answers a question of Ein and Lazarsfeld in the case of curves.