The Maximal Rank Conjecture for sections of curves

Abstract

Let C be a general curve of genus g, embedded in P r via a general linear series of degree d. In this paper, we prove the Maximal Rank Conjecture, which determines the Hilbert function of C ⊂ P r . DegenerationsIn this section, we outline the degenerations we shall use in the proof of Theorem 1.3.

The generality of a section of a curve

“…(3) For Theorem 1.6, it suffices to consider the cases where (d, g) is one of: (9, 5), (10,6), (11,7), (12,9), (16, 15), (17, 16), (18, 17).…”

“…Our three main theorems (five counting the first two cases which are trivial) give a complete description of this intersection in all of these cases. Both the results and the techniques developed here play a critical role in the author's proof of the Maximal Rank Conjecture [10], as explained in [11]. (5,2), (6,2), (6,4), (7,5), (8,6)}.…”

The Strong Maximal Rank conjecture and higher rank Brill–Noether theory

“…Proof In both cases, the multiplication map fails to be surjective and hence its dual fails to be injective. Case (1) follows from the classical Maximal Rank Conjecture, which is now a theorem of Eric Larson; see [20]. Larson's theorem implies that there exists a non‐zero extension in with , in which case and hence .…”

Good components of curves in projective spaces outside the Brill–Noether range

“…We prove that it has the expected number of moduli in the sense of Sernesi [30] (Proposition 3.1). We also prove some statements on the intersection of some elements of A(d, g; n) with a hyperplane (in the spirit of [2,4,8,10,23,26]) (see section 4). The last 5 sections are devoted to the proof of Theorem 1.1, the last one containing the numerical lemmas used in the proof.…”