3:45: Hilbert's 14th Problem and Cox RingsAna-Maria CastravetWe give a description of the generators of the total coordinate ring of the blow-up of a projective space in any number of points that lie on a rational normal curve. As a corollary we show that the algebra of invariants of the action of a two-dimensional additive group introduced by Nagata is finitely generated by certain explicit determinants. Our methods also prove that the total coordinate ring of {bar M0,6}, the moduli space of stable, 6-pointed rational curves is finitely generated. We investigate the similar question for {bar M0,n} and we conjecture that certain classes of divisors on {bar M0,n} are vertices of the effective cone. |
PDF notes, courtesy of Bjorn Poonen
5:00: The Multiplicity ConjectureDaniel ErmanA homogeneous ideal I in a polynomial ring R=k[x0, ..., xn] determines a projective scheme and also a homogeneous coordinate ring R/I. Since R/I is graded, we can take the minimal free resolution of R/I, and each slot of this resolution will be a direct sum of twists of the polynomial ring: \oplus R(-ai). The multiplicity conjecture of Herzog, Huneke, and Srinivasan states that if you look only at the largest twist from each slot, you can find an upper bound for the degree of X; a similar formula provides a conjectural lower bound. Despite intense efforts over the past couple decades, progress on the multiplicity conjecture has been slow. In this talk, I'll explain why the multiplicity conjecture has been so elusive, outline the progress that has been made, and look at a recent fresh approach to the problem. |
PDF notes, courtesy of Bjorn Poonen