# The Beilinson Complex and Canonical Rings of Irregular Surfaces

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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 $\mathbb{P}(1,1,2,3)$.