Initial ideals of weighted forms and the genus of locally Cohen-Macaulay curves

TL;DR

This paper proves the maximum genus problem for certain curves in projective 3-space by analyzing initial ideals of weighted forms, confirming a conjecture in algebraic geometry.

Contribution

It establishes the maximum genus prediction for locally Cohen-Macaulay curves when specific degree conditions are met, using initial ideals in a non-standard grading.

Findings

Confirmed the maximum genus prediction for d=s and d≥2s-1.

Proved a conjecture relating initial ideals of weighted forms to the genus of curves.

Connected algebraic properties of initial ideals to geometric genus bounds.

Abstract

Let C be a locally Cohen-Macaulay curve in complex projective 3-space. The maximum genus problem predicts the largest possible arithmetic genus g(d,s) that C can achieve assuming that it has degree d and does not lie on surfaces of degree less than s. In this paper, we prove that this prediction is correct when d=s or d is at least 2s-1. We obtain this result by proving another conjecture, by Beorchia, Lella, and the second author, about initial ideals associated to certain homogeneous forms in a non-standard graded polynomial ring.