3:45: Generic projections of projective varietiesRoya BeheshtiLet X be a nonsingular projective variety of dimension n over an algebraically closed field, and consider an embedding of X in a projective space. I will talk about generic projections of X into the projective (n+1)-space, a conjecture on the regularity of the fibers of generic projections, and some results in the direction of the conjecture. This is joint work with D. Eisenbud. |
PDF notes, courtesy of Bjorn Poonen
5:00: Matrix pencils, 0-regular sheaves, and rational normal scrollsEdward CarterA matrix pencil is an object of the form λ A+μ B, where A and B are both m × n matrices. We think of λ and μ as the variables over P1C. Under the equivalence relation of multiplication by arbitrary invertible matrices with constant entries on either side, every matrix pencil has a canonical form similar to the Jordan canonical form for square matrices. Under the same equivalence relation, there are symmetric (resp. skew-symmetric) canonical forms for symmetric (resp. skew-symmetric) pencils. Using this canonical form, we can classify 0-regular sheaves on P1.An m × n matrix pencil is a map from kn to km Ä k2, which is then an element of km Ä k2 Ä kn. In this way, we can allow the variables over P1 to exchange roles with the rows or columns of the pencil and associate each pencil to a subscheme of some rational normal scroll. |
PDF notes, courtesy of Bjorn Poonen