The Maximal Rank Conjecture for Sections of Curves

TL;DR

This paper proves a maximal rank property for hyperplane sections of unions of general curves, extending the Maximal Rank Conjecture and aiding in its proof, with specific exceptions for quadratic cases.

Contribution

It establishes the maximal rank property for hyperplane sections of unions of general curves, a key step towards proving the Maximal Rank Conjecture.

Findings

Maximal rank holds for hyperplane sections of unions of general curves.

Exceptions occur when m=2, indicating special quadratic cases.

Results support the proof of the Maximal Rank Conjecture.

Abstract

Let be a general curve of genus g embedded via a general linear series of degree d in P^r. The well-known Maximal Rank Conjecture asserts that the restriction maps H^0(O_{P^r}(m)) \to H^0(O_C(m) are of maximal rank; if known, this conjecture would determine the Hilbert function of C. In this paper, we prove an analogous statement for the hyperplane sections of unions general curves. More specifically, if H is a general hyperplane, we show that H^0(O_H(m)) \to H^0(O_{(C_1 \cup C_2 \cup \cdots \cup C_n) \cap H}(m)) is of maximal rank, except for some counterexamples when m = 2. As explained in arXiv:1809.05980, this result plays a key role in the author's proof of the Maximal Rank Conjecture.