Trygve Johnsen

Rational curves of degree at most 9 on a general quintic threefold

ABSTRACT. We prove the following form of the Clemens conjecture in low degree. Let d ≤ 9, and let F be a general quintic threefold in P 4. Then (1) the Hilbert scheme of rational, smooth and irreducible curves of degree d on F is finite, nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (−1) ⊕ (−1), and in P 4 with maximal rank. (2) On F, there are no rational, singular, reduced and irreducible curves of degree d, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3) On F, there are no connected, reduced and reducible curves of degree d with rational components.

Combinatorial Aspects of Commutative Algebra and Algebraic Geometry

We describe the positive cone generated by bigraded Betti diagrams of artinian modules of codimension two, whose resolutions become pure of a given type when taking total degrees. If the differences of these total degrees, p and q, are relatively prime, the extremal rays are parametrised by order ideals in N contained in the region px + qy < (p − 1)(q − 1). We also consider some examples concerning artinian modules of codimension three.The Cone of Betti Diagrams of Bigraded Artinian Modules of Codimension Two: M.Boij, G.Floystad.- Koszul Cycles: W.Bruns, A.Conca, T.Romer.- Boij-Soderberg Theory: D.Eisenbud, F.-O.Schreyer.- Powers of Componentwise Linear Ideals: J.Herzog, T.Hibi, H.Ohsugi.- Modules With 1-Dimensional Socle and Components of Lusztig Quiver Varieties in Type A.: J.Kamnitzer, C.Sadanand.- Realization Spaces for Tropical Fans: E.Katz, S.Payne.- A Relation Between Symmetric Polynomials and the Algebra of Classes, Motivated by Equivariant Schubert Calculus: D.Laksov.- Theory and Applications of Lattice Point Methods for Binomial Ideals: E.Miller.- Equations Defining Secant Varieties: Geometry and Computation: J.Sidman, P. Vermeire.

Wei-type duality theorems for matroids

We present several fundamental duality theorems for matroids and more general combinatorial structures. As a special case, these results show that the maximal cardinalities of fixed-ranked sets of a matroid determine the corresponding maximal cardinalities of the dual matroid. Our main results are applied to perfect matroid designs, graphs, transversals, and linear codes over division rings, in each case yielding a duality theorem for the respective class of objects.

Isolated rational curves on K3-fibered Calabi–Yau threefolds

Abstract:In this paper we study 16 complete intersection K3-fibered Calabi--Yau variety types in biprojective space ℙn1}×ℙ1. These are all the CICY-types that are K3 fibered by the projection on the second factor. We prove existence of isolated rational curves of bidegree (d,0) for every positive integer d on a general Calabi–Yau variety of these types. The proof depends heavily on existence theorems for curves on K3-surfaces proved by S. Mori and K. Oguiso. Some of these varieties are related to Calabi–Yau varieties in projective space by a determinantal contraction, and we use this to prove existence of rational curves of every degree for a general Calabi–Yau variety in projective space.

HAMMING WEIGHTS AND BETTI NUMBERS OF STANLEY-REISNER RINGS ASSOCIATED TO MATROIDS

To each linear code

Scroll codes

C

A determination of the parameters of a large class of Goppa codes

over a finite field we associate the matroid