Syzygies of Special Varieties

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9403703 Weyman The aim of this research is to study the varieties arising as the collapsing of vector bundles. Several parts of the research correspond to the most interesting examples of such varieties. The first part is to study degrees, determinantal expressions and singularities of generalized discriminants and resultants. Special attention will be paid to the classical discriminants and resultants as well as the hyperdeterminants- the discriminants of multidimensional matrices. The second part is concerned with singularities of conjugacy classes of nilpotent elements in classical Lie algebras. The main goal here is to prove the principal investigator's conjecture on non-normal orbits. The third part is to sutdy the varieties of binary forms with multiple roots. This research is concerned with a number of questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of families of polynomial equations. One can either study the geometry of the solution set or approach problems algebraically by investigating certain functions on the solution set that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry.