Isotropic Subspaces of Polylinear Forms

The dimensions of maximal isotropic subspaces of symmetric and skew-symmetric forms of degree d in general position are found.     

Invariant linear connections on homogeneous symplectic varieties

We find all homogeneous symplectic varieties of connected semisimple Lie groups that admit an invariant linear connection.The research was supported by the grant INTAS-OPEN-97-1570 of the INTAS Foundation.     

Generic algebras

The paper studies generic commutative and anticommutative algebras of a fixed dimension, their invariants, covariants and algebraic properties (e.g., the structure of subalgebras). In the case of 4-dimensional anticommutative algebras a construction is given that links the associated cubic surface and the 27 lines on it with the structure of subalgebras of the algebra. The rationality of the corresponding moduli variety is proved. In the case of 3-dimensional commutative algebras a new proof of a recent theorem of Katsylo and Mikhailov about the 28 bitangents to the associated plane quartic is given. The research was supported by Grant # MQZ300 from the ISF and Russian Government.     

Stable pair,tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces

We give a functorial normal crossing compactification of the moduli space of smooth cubic surfaces entirely analogous to the Grothendieck-Knudsen compactification .