# Singular Curves of Low Degree and Multifiltrations from Osculating Spaces

## TL;DR

This paper introduces multifiltrations from osculating spaces to analyze projections of smooth curves, classifies singularities for certain rational curves, and reestablishes a Castelnuovo bound for curves in projective space.

## Contribution

It develops a new framework using multifiltrations to study curve projections and singularities, providing classifications and a novel proof of a classical bound.

## Key findings

- Classified singularities of projected smooth curves in low-dimensional spaces.
- Determined all singularity configurations for certain rational curves.
- Reproved a case of the Castelnuovo bound using multifiltrations.

## Abstract

In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree d rational curves in $\mathbb{P}^n$ when $d - n \leq 3$ and $d < 2n$. Along the way, we describe the Schubert cycles giving rise to these projections. We also reprove a special case of the Castelnuovo bound using these multifiltrations: under the assumption $d < 2n$, the arithmetic genus of any nondegenerate degree $d$ curve in $\mathbb{P}^n$ is at most $d - n$.