KKP conjecture for minimal adjoint orbits

We prove that LG models for minimal semisimple adjoint orbits satisfy the Katzarkov--Kontsevich--Pantev conjecture about new Hodge theoretical invariants.Comment: 22 page

Examples of differential geometric behaviour of projective varieties in positive characteristic

Here we study three examples of differential geometric behaviour of projective varieties in positive characteristic: (1) the classification of smooth surfaces in P2n+1 whose m-th osculating spaces have everywhere dimension 2m (1 ≤ m ≤ n); (2) hypersurfaces with Hessian rank 0; (3) singular hypersurfaces in weighted projective spaces whose tangent sheaf is locally free and a subbundle of the restricted tangent bundle

Curve classes on irreducible holomorphic symplectic varieties

We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3 type and of Generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.Comment: 15 page

Four lectures on secant varieties

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in Mathematics & Statistics), Springer/Birkhause

The Degree and regularity of vanishing ideals of algebraic toric sets over finite fields

Let X* be a subset of an affine space A^s, over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x --> [x] and x --> [(x,1)] respectively, where [x] and [(x,1)] are points in the projective spaces P^{s-1} and P^s respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud-Goto regularity conjecture. We give optimal bounds for the regularity when the graph is bipartite. It is shown that X* is an affine torus if and only if I(Y) is a complete intersection. We present some applications to coding theory and show some bounds for the minimum distance of parameterized linear codes for connected bipartite graphs