Interpolation for Curves in Projective Space with Bounded Error

TL;DR

This paper establishes a bounded-error criterion for the existence of curves passing through general points in projective space, advancing understanding of special linear series and their interpolation properties, with implications for the Maximal Rank Conjecture.

Contribution

It proves a bounded-error analog for the existence of curves with given degree and genus passing through points, extending previous results to special linear series and normal bundle interpolation.

Findings

Existence of curves with n points under a stricter inequality, with a -3 bound.

Normal bundle N_C(-1) satisfies interpolation for large degree curves.

Lower bounds on the number of points in hyperplane sections of general curves.

Abstract

Given n general points p_1, p_2,..., p_n \in P^r, it is natural to ask whether there is a curve of given degree d and genus g passing through them; by counting dimensions a natural conjecture is that such a curve exists if and only if \[n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor.\] In the case of curves with nonspecial hyperplane section, the above conjecture was recently shown to hold with exactly three exceptions. In this paper, we prove a "bounded-error analog" for special linear series on general curves; more precisely we show that existance of such a curve subject to the stronger inequality \[n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor - 3.\] Note that the -3 cannot be replaced with -2 without introducing exceptions (as a canonical curve in P^3 can only pass through 9 general points, while a naive dimension count predicts 12). We also use the same technique to prove that the twist of the normal bundle N_C(-1) satisfies interpolation for curves whose degree is sufficiently large relative to their genus, and deduce from this that the number of general points contained in the hyperplane section of a general curve is at least \[\min\left(d, \frac{(r - 1)^2 d - (r - 2)^2 g - (2r^2 - 5r + 12)}{(r - 2)^2}\right).\] As explained in arXiv:1809.05980, these results play a key role in the author's proof of the Maximal Rank Conjecture.