Interpolation for Brill-Noether curves in $\mathbb{P}^4$

TL;DR

This paper calculates the number of general points a typical Brill-Noether curve in four-dimensional projective space passes through and extends the result to points constrained in a hyperplane, aiding in proving the Maximal Rank Conjecture.

Contribution

It provides new enumerative results for Brill-Noether curves in P^4, including cases with hyperplane constraints, which are crucial for the Maximal Rank Conjecture proof.

Findings

Number of general points a Brill-Noether curve passes through in P^4

Extension to points constrained in a hyperplane

Supports proof of the Maximal Rank Conjecture

Abstract

In this paper, we compute the number of general points through which a general Brill-Noether curve in $\mathbb{P}^4$ passes. We also prove an analogous theorem when some points are constrained to lie in a transverse hyperplane. As explained in arXiv:1809.05980, these results play an essential role in the first author's proof of the Maximal Rank Conjecture.