2014 MINIMAL RESOLUTIONS, CHOW FORMS AND ULRICH BUNDLES ON K 3 SURFACES

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 that for 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 curves ,