2020 CMS Winter Meeting

Montreal, December 4 - 7, 2020


Derived Categories and (Non)commutative Algebraic Geometry
Org: Matthew Ballard (USC), Nitin Chidambaram (Alberta) and David Favero (Alberta)




COLIN INGALLS, Carleton University
Explicit coverings of families of elliptic surfaces by squares of curves  [PDF]

We show that, for each $n>0,$ there is a family of elliptic surfaces which are covered by the square of a curve of genus $2n+1,$ and whose Hodge structures have an action by $\mathbb{Q}(\sqrt{-n})$. By considering the case n=3, we show that one particular family of K3 surfaces are covered by the square of genus 7. Using this, we construct a correspondence between the square of a curve of genus 7 and a general K3 surface in $\mathbb{P}^4$ with 15 ordinary double points up to isogeny. This gives an explicit proof of the Kuga-Satake-Deligne correspondence for these K3 surfaces and any K3 surfaces isogenous to them, and further, a proof of the Hodge conjecture for the squares of these surfaces. We conclude that the motives of these surfaces are Kimura-finite. Our analysis gives a birational equivalence between a moduli space of curves with additional data and the moduli space of these K3 surfaces with a specific elliptic fibration. This is joint work with Adam Logan and Owen Patashnick.

ELLEN KIRKMAN, Wake Forest University
Degree bounds for Hopf actions on Artin-Schelter regular algebras  [PDF]

In 1915 E. Noether proved that for a field $\Bbbk$ of characteristic zero and a finite group $G$ acting naturally on a polynomial ring $\Bbbk[x_1, \dots, x_n]$, the degrees of minimal generators of the subring of invariants are bounded above by the order of the group. In 2011, using Castelnuovo-Mumford regularity, P. Symonds proved that for a general field $\Bbbk$, an upper bound is $n(|G| -1)$ when $n \geq 2$ and $|G|>1$. Replacing $\Bbbk[x_1, \dots, x_n]$ by an Artin-Schelter regular algebra $A$ and $G$ by a semisimple Hopf algebra $H$, we prove analogues of results of Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin and Symonds on bounds on the degrees of generators of the subring of invariants and on the degrees of syzygies of modules over the invariant subring. We further explore Castelnuovo-Mumford regularity and related weighted sums of homological and internal degrees in complexes of graded $A$-modules for noncommutative algebras. This is joint work with Robert Won and James J. Zhang.


MAX LIEBLICH, Washington


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