Greedy weights for matroids

We introduce greedy weights of matroids, inspired by those for linear codes. We show that a Wei duality holds for two of these types of greedy weights for matroids. Moreover we show that in the cases where the matroids involved are associated to linear codes, our definitions coincide with those for codes. Thus our Wei duality is a generalization of that for linear codes given by Schaathun. In the last part of the paper we show how some important chains of cycles of the matroids appearing, correspond to chains of component maps of minimal resolutions of the independence complex of the corresponding matroids. We also relate properties of these resolutions to chainedness and greedy weights of the matroids, and in many cases codes, that appear.

Crepant resolutions and open strings

In the present paper, we formulate a Crepant Resolution Correspondence for open Gromov–Witten invariants (OCRC) of toric Lagrangian branes inside Calabi–Yau 3-orbifolds by encoding the open theories into sections of Givental’s symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan–Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates–Corti–Iritani–Tseng and Ruan, we furthermore propose (1) a general form of the morphism entering the OCRC, which arises from a geometric correspondence between equivariant K-groups, and (2) an all-genus version of the OCRC for Hard Lefschetz targets. We provide a complete proof of both statements in the case of minimal resolutions of threefold {A_{n}}-singularities; as a necessary step of the proof we establish the all-genus closed Crepant Resolution Conjecture with descendents in its strongest form for this class of examples. Our methods rely on a new description of the quantum D-modules underlying the equivariant Gromov–Witten theory of this family of targets.

A smörgåsbord of scalar-flat Kähler ALE surfaces

There are many known examples of scalar-flat Kähler ALE surfaces, all of which have group at infinity either cyclic or contained in {{\rm{SU}}(2)} . The main result in this paper shows that for any non-cyclic finite subgroup Γ \subset U(2) containing no complex reflections, there exist scalar-flat Kähler ALE metrics on the minimal resolution of \mathbb{C}^{2} /Γ, for which Γ occurs as the group at infinity. Furthermore, we show that these metrics admit a holomorphic isometric circle action. It is also shown that there exist scalar-flat Kähler ALE metrics with respect to some small deformations of complex structure of the minimal resolution. Lastly, we show the existence of extremal Kähler metrics admitting holomorphic isometric circle actions in certain Kähler classes on the complex analytic compactifications of the minimal resolutions.

Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces

The Minimal Resolution Conjecture (MRC) for points on a projective variety

COX RINGS OF MINIMAL RESOLUTIONS OF SURFACE QUOTIENT SINGULARITIES

We investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety.