The Mercat Conjecture for stable rank 2 vector bundles on generic curves

We prove the Mercat Conjecture for rank 2 vector bundles on generic curves of every genus. For odd genus, we identify the effective divisor in Mg where the Mercat Conjecture fails and compute its slope. The Clifford index of a smooth curve C, defined as the non-negative quantity Cliff(C) := min { deg(L)− 2r(L) : L ∈ Pic(C), h(C,L) ≥ 2, deg(L) ≤ g − 1 } stratifies the moduli space Mg of curves of genus g, with the smallest stratum being that consisting of hyperelliptic curves, which have Clifford index zero. While the definition is obviously inspired by the classical Clifford theorem, the notion came to real prominence in the works of Green, Lazarsfeld [G], [GL], Voisin [V2] and many others in the context of syzygies of canonical curves. For a general curve C of genus g, one has Cliff(C) = bg−1 2 c; the locus of curves [C] ∈Mg with Cliff(C) < bg−1 2 c is a subvariety of codimension 1 (respectively 2) when g is odd (respectively even) much studied by Harris and Mumford [HM]. It has been a long standing problem to find an adequate definition of the Clifford index for higher rank vector bundles on curves. A higher rank Clifford index should not only capture the behavior of the generic curve from the point of view of special higher rank vector bundles, but also provide a geometrically meaningful stratification ofMg. An interesting definition has been put forward by Lange and Newstead [LN1]. For a semistable vector bundle E of rank 2 and slope μ(E) on a curve C of genus g, one defines its Clifford index as Cliff(E) := μ(E)− h(C,E) + 2 ≥ 0. The rank 2 Clifford index of C is then defined as the quantity Cliff2(C) := min { Cliff(E) : E ∈ UC(2, d), d ≤ 2g − 2, h(C,E) ≥ 4 } . Observe that if L is a line bundle on C with deg(L) − 2h0(C,L) + 2 = Cliff(C) (that is, L computes the Clifford index of C), then Cliff(L ⊕ L) = Cliff(C). In particular, the inequality Cliff2(C) ≤ Cliff(C) holds for every curve C of genus g. The main achievement of this paper is the proof of the following result: Theorem 1. For a general curve C of genus g the following equality holds Cliff2(C) = Cliff(C) = ⌊g − 1 2 ⌋ . Equivalently, if E is a semistable rank 2 vector bundle on C contributing to Cliff2(C), then h(C,E) ≤ μ(E) + 2− Cliff(C). Mercat [Me] conjectured the equality Cliff2(C) = Cliff(C) for every smooth curve [C] ∈Mg. Counterexamples to this expectation have been found using Noether-Lefschetz special K3 surfaces in [FO1], [FO2] and [LN2]. However these examples are special in moduli and Theorem 1 proves Mercat’s Conjecture for generic curves of every genus.