The Generality of a Section of a Curve

TL;DR

This paper demonstrates that for a broad class of general curves mapped into projective space, their intersections with general hypersurfaces produce a generic set of points, with only finitely many exceptions, advancing understanding in algebraic geometry.

Contribution

It establishes the generality of intersection points between general curves and hypersurfaces across various dimensions, with finitely many exceptions, supporting the Maximal Rank Conjecture.

Findings

Intersections of general curves with quadrics yield general point collections, except for six cases.

Similar results hold for intersections with other hypersurfaces across different dimensions.

Supports the proof of the Maximal Rank Conjecture.

Abstract

Let f: C --> P^3 be a general curve of genus g, mapped to P^3 via a general linear series of degree d; and let Q be a general (and thus smooth) quadric. In this paper, we show that the points of intersection f(C) \cap Q give a general collection of 2d points on Q, except for exactly six exceptional cases. We also prove similar theorems for every other pair (r, n) for which, except for only finitely many pairs (d, g), the intersection of a general curve of genus g mapped to P^r via a general linear series of degree d, with a general hypersurface S of degree n, is a general collection of dn points on S. As explained in arXiv:1809.05980, these results play a key role in the author's proof of the Maximal Rank Conjecture