On Parameterizations of plane rational curves and their syzygies

TL;DR

This paper characterizes the splitting types of plane rational curves and links them to geometric projections from rational normal surfaces, providing a geometric criterion for specific syzygy configurations.

Contribution

It establishes a geometric characterization of the splitting type (a,b) for plane rational curves in terms of projections from rational normal surfaces.

Findings

Splitting type (a,b) = (k, d-k) occurs if and only if the curve is a projection of a rational curve on a rational normal surface.

Provides a geometric description linking syzygies of parameterizations to projections from higher-dimensional surfaces.

Characterizes when specific syzygy configurations occur based on the curve's geometric origin.

Abstract

Let $C$ be a plane rational curve of degree $d$ and $p:\tilde C \rightarrow C$ its normalization. We are interested in the splitting type $(a,b)$ of $C$, where $\mathcal{O}_{\mathbb{P}^1}(-a-d)\oplus \mathcal{O}_{\mathbb{P}^1}(-b-d)$ gives the syzigies of the ideal $(f_0,f_1,f_2)\subset K[s,t]$, and $(f_0,f_1,f_2)$ is a parameterization of $C$. We want to describe in which cases $(a,b)=(k,d-k)$ ($2k\leq d)$, via a geometric description; namely we show that $(a,b)=(k,d-k)$ if and only if $C$ is the projection of a rational curve on a rational normal surface in $\mathbb{P}^{k+1}$.