3:00pm - 5:00pmComputations in algebraic geometry
Chair(s): Diane Maclagan (University of Warwick), Gregory G. Smith (Queen's University)
This minisymposium highlights the use of computation inside algebraic geometry. Computations enter algebraic geometry in several different ways including numerical strategies, symbolic calculations, experimentation, and simply as a fundamental conceptual tool. Our speakers will showcase many of these aspects together with some applications.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Regularity of S_n-invariant monomial ideals
Claudiu Raicu
University of Notre Dame
Consider a polynomial ring R in n variables with the action of the symmetric group Sn by coordinate permutations. I will describe an explicit recipe for computing the graded components of the modules Ext(I,R), when I is an arbitrary Sn-invariant monomial ideal, as well as the maps induced by inclusions of such ideals. As a consequence, this gives explicit formulas for the the regularity of Sn-invariant monomial ideals.
A homological approach to numerical Godeaux surfaces
Wolfram Decker
University of Kaiserslautern
Numerical Godeaux surfaces provide the first case in the geography of minimal surfaces of general type. By work of Miyaoka and Reid it is known that the torsion group of such a surface is cyclic of order at most 5, a full classification has been given for the cases where this order is 3,4, or 5. In my talk, I will discuss recent progress by Isabel Stenger towards the classification of numerical Godeaux surfaces with a trivial torsion group. Following a suggestion by Frank-Olaf Schreyer, the starting point of Stenger's work is a syzygy-type approach to the study of the canonical ring of such a surface. Particular attention is paid to the hyperelliptic curves arising in the fibration induced by the bicanonical system.
Asymptotic syzygies for products of projective space
Juliette Bruce
University of Wisconsin
We will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.
Where can toric syzygies live?
Milena Hering
University of Edinburgh
Syzygies of toric varieties admit a natural grading by the character lattice of the corresponding torus. I will give some results on the the regions in the character lattice in which toric syzygies can be supported. This is joint work with Castryck and Lemmens.