The Connectedness of Space Curve Invariants

TL;DR

This paper extends the connectedness property of invariants from points in the projective plane to space curves in three-dimensional projective space, providing new restrictions on their algebraic and geometric properties.

Contribution

It proves that invariants of irreducible, non-degenerate space curves in three-dimensional projective space are connected, generalizing prior results for points in the plane.

Findings

Invariants of space curves satisfy a connectedness property.

This restricts possible generic initial ideals of such curves.

Provides new insights into their Hilbert functions.

Abstract

It is a result of Gruson and Peskine that the invariants of a set points in $\ptwo$ in general position are connected. Associated to a space curve there are sequences of invariants which generalize the invariants of points in $\ptwo$. The main result of this paper is to show that the invariants of reduced, irreducible, non-degenerate curves in $\pthree$ also satisfy a connectedness property. This result greatly restricts the kinds of Borel-fixed monomial ideals which can occur as generic initial ideals of such curves and thus gives us more control over their Hilbert functions.