The Minimal Resolution Conjecture (MRC) for points on a projective variety X ⊂ P r predicts that the minimal graded free resolution of a general set Γ ⊂ X of points is as simple as the geometry of X allows. Originally, the most studied case has been that when X = P r , see [EPSW]. The general form of the MRC for subvarieties X ⊂ P r was formulated in [Mus] and [FMP]. The Betti diagram of a large enough set Γ ⊂ X consisting of γ general points is obtained from the Betti diagram of X, by adding two rows, indexed by u − 1 and u, where u is an integer depending on γ. All differences b i+1,u−1 (Γ) − b i,u (Γ) are known and depend on the Hilbert polynomial P X and i, u and γ, see [FMP]. The Minimal Resolution Conjecture for γ general points on X predicts thatfor each i ≥ 0, in which case, the Betti numbers of Γ are explicitly given in terms of P X and γ. The Ideal Generation Conjecture (IGC) predicts the same vanishing but only for i = 1, that is, b 2,u−1 (Γ) · b 1,u (Γ) = 0; equivalently, the number of generators of the ideal I Γ /I X is minimal.In [FMP], the Minimal Resolution Conjecture for points on curves is reformulated in geometric terms. For a globally generated linear series ℓ = (L, V ) ∈ G r d (C), we consider the kernel vector bundle M V defined via the evaluation sequenceThen MRC holds for C |V | ֒→ P r if and only if M V satisfies the Raynaud property (R)for each i = 0, . . . , r and a general line bundle ξ on C with deg(ξ)in which case we refer to C ⊂ P r as being a curve of integer slope), property (R) is satisfied if and only if for i = 0, . . . , r, the cycleOur first result is a proof of MRC for curves C ⊂ P r of integer slope µ := d r ∈ Z ≥1 . Theorem 0.1. The Minimal Resolution Conjecture holds for a general embedding C ֒→ P r of degree µr of any curve C with general moduli and for any integers µ, r ≥ 1. The hypothesis on the generality of C implies that its genus g satisfies the inequality g ≤ (r + 1)(µ − 1) imposed by Brill-Noether theory. We have similarly complete results for curves C ⊂ P r of degree d ≡ ±1 mod r, see Theorem 1.6.In the case of curves C |L| ֒→ P d−g embedded by a complete linear system of degree d ≥ 2g + 5, counterexamples to MRC for points on C were found in [FMP] 1 ; observe that in these cases µOn the other hand, MRC holds for all smooth canonical curves C ⊂ P g−1 , see [FMP], as well as for general line bundles of degree 2g, see [B1]. In both these cases, one has µ = 2. This confusing state of affairs is reminiscent of the situation for the projective space P r , where it is known [HS] that MRC holds for r ≥ 4 and γ very large with respect to r, but fails for each r ≥ 6, r = 9 for many values of γ, see [EPSW]. Our next result show that for curves, independently of the genus, the Clifford line line d = 2r in the (d, r)-plane governs whether MRC holds for a general curve C ⊂ P r of genus g and degree d. Note that no assumption is made regarding the completeness of the linear series (L, V ) inducing the map ϕ V : C ֒→ P r . Inequality (2) in Theorem 0.2 is satisfied when W...

We describe applications of Koszul cohomology to the BrillNoether theory of rank 2 vector bundles. Among other things, we show that in every genus g > 10, there exist curves invalidating Mercat's Conjecture for rank 2 bundles. On the other hand, we prove that Mercat's Conjecture holds for general curves of bounded genus, and its failure locus is a Koszul divisor in the moduli space of curves. We also formulate a conjecture concerning the minimality of Betti diagrams of suitably general curves, and point out its consequences to rank 2 Brill-Noether theory.

The Prym map of type (g, n, r ) associates to every cyclic covering of degree n of a curve of genus g ramified at a reduced divisor of degree r the corresponding Prym variety. We show that the corresponding map of moduli spaces is generically finite in most cases. From this we deduce the dimension of the image of the Prym map.

We discuss the role of K3 surfaces in the context of Mercat's conjecture in higher rank Brill-Noether theory. Using liftings of Koszul classes, we show that Mercat's conjecture in rank 2 fails for any number of sections and for any gonality stratum along a Noether-Lefschetz divisor inside the locus of curves lying on K3 surfaces. Then we show that Mercat's conjecture in rank 3 fails even for curves lying on K3 surfaces with Picard number 1. Finally, we provide a detailed proof of Mercat's conjecture in rank 2 for general curves of genus 11, and describe explicitly the action of the Fourier-Mukai involution on the moduli space of curves.

We investigate the number and the geometry of smooth hyperelliptic curves on a general complex abelian surface. We show that the only possibilities of genera of such curves are 2, 3, 4 and 5. We focus on the genus 5 case. We prove that up to translation, there is a unique hyperelliptic curve in the linear system of a general (1, 4) polarised abelian surface. Moreover, the curve is invariant with respect to a subgroup of translations isomorphic to the Klein group. We give the decomposition of the Jacobian of such a curve into abelian subvarieties displaying Jacobians of quotient curves and Prym varieties. Motivated by the construction, we prove the statement: everyétale Klein covering of a hyperelliptic curve is a hyperelliptic curve, provided that the group of 2-torsion points defining the covering is non-isotropic with respect to the Weil pairing and every element of this group can be written as a difference of two Weierstrass points.