Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces

Abstract

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...

Ulrich bundles on ruled surfaces

“…The existence of a special Ulrich bundle implies that the associated Cayley-Chow form is represented as a linear pfaffian [ES03]. Recently, this problem has been solved for general K3 surfaces [AFO12], [CKM12], for del Pezzo surfaces in [MP2], for ACM rational surfaces in [MP] and for surfaces with q = p g = 0, [Bea16]. We will focus here on surfaces with Kodaira dimension −∞.…”

Ulrich line bundles on Enriques surfaces with a polarization of degree four

“…A K3 surface whose Picard group is generated by H S automatically satisfies this hypothesis. In [AFO12], H S was supposed to be very ample. However, the exactly same proof goes through even if we only assume that H S is ample and globally generated.…”

“…As noted in [AFO12], the sufficient Brill-Noether condition on K3 surfaces is used only to ensure the existence of a base-point-free pencil of degree 5 2 H 2 S + 4 on the cubic sections. However, there are cases not covered by this Brill-Noether condition and which still carry Ulrich bundles, and even special Ulrich bundles.…”

Examples of surfaces which are Ulrich–wild

“…At present, answers to the questions above are known in a number of particular cases: e.g., see [2], [4], [6], [7], [12], [13], [14], [18], [19], [20], [35], [36], [37], [41].…”