Strata of rational space curves

TL;DR

This paper characterizes the structure and stratification of rational space curves based on their mu-invariant, providing explicit formulas and decompositions related to parameterizations and their geometric properties.

Contribution

It introduces a detailed description of mu-strata, including the closure, codimension of non-proper parametrizations, and a decomposition based on rational normal scrolls.

Findings

Explicit formula for codimension of non-proper parametrizations.

Description of the closure of mu-strata.

Decomposition of the smallest mu-stratum by rational normal scrolls.

Abstract

The mu-invariant mu = (\mu_1,\mu_2,\mu_3) of a rational space curve gives important information about the curve. In this paper, we describe the structure of all parameterizations that have the same mu-type, what we call a mu-stratum, and as well the closure of strata. Many of our results are based on papers by the second author that appeared in the commutative algebra literature. We also present new results not in the earlier papers, including an explicit formula for the codimension of the locus of non-proper parametrizations within each mu-stratum and a decomposition of the smallest mu-stratum based on which two-dimensional rational normal scroll the curve lies on.