Mathematical Sciences: Topics in Algebraic Geometry

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EIN Ein proposes to investigate the following and related problems in algebraic geometry. (1) Linear systems on higher dimensional varieties. One of the basic tools in investigating the geometry of an algebraic variety is to study the properties of the linear systems on the variety. Ein (jointly with R. Lazarsfeld) proved a conjecture of Fujita on the freeness of a large class of adjoint linear systems on threefolds. They would like to apply the new techniques developed in studying this conjecture to study singular divisors and the geometry of the special subvarieties. In particular they plan to study the singularities of the theta divisors on abelian varieties. They would also like to extend their results on linear systems to higher dimensions and to study questions on very ampleness. (2) Local positivity of ample line bundles and Seshadri constants. The Seshadri constant is an important invariant that measures the local positivity of an ample line bundle. Ein would like to work on the conjecture that the Seshadri constant is greater or equal to one for a general point x in our projective variety. (3) Syzygies of algebraic varieties. Ein would like to investigate the relation between the geometry of the variety and the syzygies of the coordinate ring of the variety while the variety is embedded by an adjoint linear system. (4) Subvarieties od generic complete intersections. Ein has proved a result that all the subvarieties of a generic complete intersection of high degree are of general type. This result has recently been improved by Voisin. Ein would like to extend the result to study the deformations of subvarieties of a generic complete intersection while the complete intersection variety is a Calabai-Yau manifold. These problems take on new significance because of their relations to physics and mirror symmetry. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In the origin, it treated figures that could be defined in the plne by the simplest equations, namesly polynomials. Nowadays, the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding applicaton in those fields as well as in physics, theoretical computer science, and robotics.