Departmental PhD Thesis Exam

Event Information Betti Tables of Maximal Cohen-Macaulay Modules over the Cones of Elliptic Normal Curves
14:10 on Thursday April 30, 2015
15:00 on Thursday April 30, 2015
BA6183, Bahen Center, 40 St. George St.
Alexander Pavlov
http://blog.math.toronto.edu/GraduateBlog/files/2015/04/ut-thesis.pdf
University of Toronto

Graded Betti numbers are classical invariants of finitely generated modules describing the shape of a minimal free resolution. We show that for maximal Cohen-Macaulay modules over a homogeneous coordinate rings of smooth Calabi-Yau varieties $X$ computation of Betti numbers can be reduced to computations of dimensions of certain Hom groups in the bounded derived category $D^b(X)$.

In the simplest case of a smooth elliptic curve $E$ we use our formula to get explicit answers for Betti numbers. Description of the automorphism group of the derived category $D^b(E)$ in terms of the spherical twist functors of Seidel and Thomas plays a major role in our approach. We study the homogeneous coordinate rings of the embeddings of the elliptic curve $E$ into projective spaces by a complete linear system of degree $n>0$.

Case $n =3$ is the simplest case of a smooth plane cubic. Here we show that there are only four possible shapes of the Betti tables up to a shifts in internal degree $(\bullet)$, and two possible shapes up to a shift in internal degree and taking syzygies.

For elliptic normal curves of degree $n>3$ recursive formulae for the Betti numbers are given, and possible shapes of the Betti tables are described. Results on the Betti numbers are applied to study Koszul and Ulrich modules over the homogeneous coordinate ring.

For $n=1,2$ the elliptic curve $E$ is embedded as a hypersurface into a weighted projective space. Homogeneous coordinate rings are known as minimal elliptic singularities $\widetilde{E_7}$ (for $n=2$) and $\widetilde{E_8}$ (for $n=1$). We show that the same approach to Betti numbers works. In fact formulae for the Betti numbers in these cases are even simpler than in the plane cubic case.

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.