Title of Seminar: Algebraic Geometry Preprint Seminar 2022
Title of Talk: Volumes of line bundles as limits on certain generically nonreduced schemes
Speaker: Suprajo Das, IIT, Bombay
Date: March 1, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: In this talk we shall discuss a recent article (arXiv$2007.12925$v$2$) by Dale Cutkosky and Roberto Nunez. Let $X$ be a proper scheme of dimension $d$ over a field $k$ and let $\mathcal{L}$ be a line bundle on $X$. The \emph{volume} of $\mathcal{L}$ is defined as $$\mathrm{vol}(\mathcal{L}):= \limsup\limits_{n\to\infty}\dfrac{\dim_k H^0(X,\mathcal{L}^{\otimes n})}{n^d/d!}.$$ A fundamental problem is to study whether the $\limsup$ in the definition of volume can be replaced by a limit. It is known to exist as a limit on generically reduced schemes. In this preprint, the authors show that the volume exists as a limit in certain classes of schemes which are not necessarily generically reduced.
Title of Talk: Volumes of line bundles as limits on certain generically nonreduced schemes
Speaker: Suprajo Das, IIT, Bombay
Date: March 1, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: In this talk we shall discuss a recent article (arXiv$2007.12925$v$2$) by Dale Cutkosky and Roberto Nunez. Let $X$ be a proper scheme of dimension $d$ over a field $k$ and let $\mathcal{L}$ be a line bundle on $X$. The \emph{volume} of $\mathcal{L}$ is defined as $$\mathrm{vol}(\mathcal{L}):= \limsup\limits_{n\to\infty}\dfrac{\dim_k H^0(X,\mathcal{L}^{\otimes n})}{n^d/d!}.$$ A fundamental problem is to study whether the $\limsup$ in the definition of volume can be replaced by a limit. It is known to exist as a limit on generically reduced schemes. In this preprint, the authors show that the volume exists as a limit in certain classes of schemes which are not necessarily generically reduced.