Abstract:

Let $X$ be a projective toric variety of dimension $n$ over an arbitrary field and let $L$ be an ample line bundle on $X$. For $k \geq 0$, it is in general difficult to determine whether $L^{\otimes k}$ is very ample and whether it additionally gives a projectively normal embedding. These two properties are equivalent to the \emph{very ampleness}, respectively \emph{normality}, of the corresponding polytope. By a result of Ewald-Wessels, both statements are classically known to hold for $k \geq n - 1$.

In this talk we study embeddings of weighted projective spaces $\mathbb{P}(a_0, \ldots, a_n)$ via their corresponding simplices. If one weight is equal to one, these are so-called rectangular simplices. In this case, we give multiple criteria (depending on arithmetic properties of the weights $a_i$) to obtain bounds for the power $k$ which are sharp in many cases. We also introduce combinatorial tools that allow us to systematically construct families exhibiting extremal behaviour. These results extend earlier work of Payne, Hering and Bruns-Gubeladze.

This is joint work with Praise Adeyemo (U Ibadan) and Dominic Bunnett (TU Berlin)