The Beilinson Complex and Canonical Rings of Irregular Surfaces

The Beilinson Complex and Canonical Rings of Irregular Surfaces Synopsis

An important theorem by Beilinson describes the bounded derived category of coherent sheaves on $\mathbb{P}^n$, yielding in particular a resolution of every coherent sheaf on $\mathbb{P}^n$ in terms of the vector bundles $\Omega_{\mathbb{P}^n}^j(j)$ for $0\le j\le n$. This theorem is here extended to weighted projective spaces. To this purpose we consider, instead of the usual category of coherent sheaves on $\mathbb{P}({\rm w})$ (the weighted projective space of weights $\rm w=({\rm w}_0,\dots,{\rm w}_n)$), a suitable category of graded coherent sheaves (the two categories are equivalent if and only if ${\rm w}_0=\cdots={\rm w}_n=1$, i.e. $\mathbb{P}({\rm w})= \mathbb{P}^n$), obtained by endowing $mathbb{P}({\rm w})$ with a natural graded structure sheaf. The resulting graded ringed space $\overline{\mathbb{P}}({\rm w})$ is an example of graded scheme (in chapter 1 graded schemes are defined and studied in some greater generality than is needed in the rest of the work).Then in chapter 2 we prove for graded coherent sheaves on $\overline{\mathbb{P}}({\rm w})$ a result which is very similar to Beilinson's theorem on $\mathbb{P}^n$, with the main difference that the resolution involves, besides $\Omega_{\overline{\mathbb{P}}({\rm w})}^j(j)$ for $0\le j\le n$, also $\mathcal{O}_{\overline{\mathbb{P}}({\rm w})}(1)$ for $n-\sum_{i=0}^n{\rm w}_i\1\0$. This weighted version of Beilinson's theorem is then applied in chapter 3 to prove a structure theorem for good birational weighted canonical projections of surfaces of general type (i.e., for morphisms, which are birational onto the image, from a minimal surface of general type $S$ into a $3$-dimensional $\mathbb{P}({\rm w})$, induced by $4$ sections $\sigma_i\in H\0(S, \mathcal{O}_S({\rm w}_iK_S))$).This is a generalization of a theorem by Catanese and Schreyer (who treated the case of projections into $\mathbb{P}^3$), and is mainly interesting for irregular surfaces, since in the regular case a similar but simpler result (due to Catanese) was already known. The theorem essentially states that giving a good birational weighted canonical projection is equivalent to giving a symmetric morphism of (graded) vector bundles on $\overline{\mathbb{P}}({\rm w})$, satisfying some suitable conditions. Such a morphism is then explicitly determined in chapter 4 for a family of surfaces with numerical invariants $p_g=q=2$, $K^2=4$, projected into $\mathbb{P}(1,1,2,3)$.