Problems in Algebraic Geometry

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Lazarsfeld will work on a number of problems in algebraic geometry. A first series of questions concerns the algebraic properties of large degree embeddings of projective varieties. The equations defining curves have been studied intensively over the last twenty-five years, but until recently little was known about the algebraic behavior of more general varieties. In this direction, Lazarsfeld will continue his work with Ein on the asymptotic structure of the syzygies of higher-dimensional varieties as the positivity of the embedding line bundle grows. Lazarsfeld will also work on a number of questions concerning the positivity properties of higher-codimension algebraic cycles. Growing out of his work with Debarre, Ein and Voisin, the idea here is to explore the higher-codimension analogues of classical notions of positivity for cycles of codimension one. Algebraic geometry, one of the oldest and most central fields of mathematics, studies geometric properties of solutions to systems of polynomial equations in several variables. It makes connections with many other fields of mathematics, ranging from number theory to topology, algebra, and complex analysis. Algebraic geometry has also found important applications to problems in such diverse areas as coding theory, theoretical physics and the mathematics of computation. In a first series of problems, Lazarsfeld will study the properties of the equations cutting out a fixed geometric locus. It is hoped that this research will open up new points of contact between the fields of algebraic geometry and commutative algebra. Lazarsfeld will also work on some conjectures concerning the manner in which smaller geometric loci can sit inside larger ones.