Algebraic matroids with graph symmetry

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a) algebraic matroids, we expose cryptomorphisms making them accessible to techniques from commutative algebra. This allows us to introduce for each circuit in an algebraic matroid an invariant called circuit polynomial, generalizing the minimal poly- nomial in classical Galois theory, and studying the matroid structure with multivariate methods. For (b) matroids with symmetries we introduce combinatorial invariants capturing structural properties of the rank function and its limit behavior, and obtain proofs which are purely combinatorial and do not assume algebraicity of the matroid; these imply and generalize known results in some specific cases where the matroid is also algebraic. These results are motivated by, and readily applicable to framework rigidity, low-rank matrix completion and determinantal varieties, which lie in the intersection of (a) and (b) where additional results can be derived. We study the corresponding matroids and their associated invariants, and for selected cases, we characterize the matroidal structure and the circuit polynomials completely

Points fattening on P^1 x P^1 and symbolic powers of bi-homogeneous ideals

We study symbolic powers of bi-homogeneous ideals of points in the Cartesian product of two projective lines and extend to this setting results on the effect of points fattening obtained by Bocci, Chiantini and Dumnicki, Szemberg, Tutaj-Gasi\'nska. We prove a Chudnovsky-type theorem for bi-homogeneous ideals and apply it to classification of configurations of points with minimal or no fattening effect. We hope that the ideas developed in this project will find further algebraic and geometric applications e.g. to study similar problems on arbitrary surfaces.Comment: 12 pages, notes from a workshop on linear series held in Lanckoron

On the existence of branched coverings between surfaces with prescribed branch data, II

For a given branched covering between closed connected surfaces, there are several easy relations one can establish between the Euler characteristics of the surfaces, their orientability, the total degree, and the local degrees at the branching points, including the classical Riemann-Hurwitz formula. These necessary relations have been khown to be also sufficient for the existence of the covering except when the base surface is the sphere (and when it is the projective plane, but this case reduces to the case of the sphere). If the base surface is the sphere, many exceptions are known to occur and the problem is widely open. Generalizing methods of Baranski, we prove in this paper that the necessary relations are actually sufficient in a specific but rather interesting situation. Namely under the assumption that the base surface is the sphere, that there are three branching points, that one of these branching points has only two preimages with one being a double point, and either that the covering surface is the sphere and that the degree is odd, or that the covering surface has genus at least one, with a single specific exception. For the case of the covering surface the sphere we also show that for each even degree there are precisely two exceptions.Comment: 38 pages, 21 figures. This is a sequel of math.GT/050843