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The Beilinson Complex and Canonical Rings of Irregular Surfaces
Alberto Canonaco, Università di Pavia, Italy
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Memoirs of the American Mathematical Society
2006; 99 pp; softcover
Volume: 183
ISBN-10: 0-8218-4193-9
ISBN-13: 978-0-8218-4193-8
List Price: US$63 Individual Members: US$37.80
Institutional Members: US\$50.40
Order Code: MEMO/183/862

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 $$\mathbb{P}(\mathrm{w})$$ (the weighted projective space of weights $$\mathrm{w}=(\mathrm{w}_0,\dots,\mathrm{w}_n)$$), a suitable category of graded coherent sheaves (the two categories are equivalent if and only if $$\mathrm{w}_0=\cdots=\mathrm{w}_n=1$$, i.e. $$\mathbb{P}(\mathrm{w})= \mathbb{P}^n$$), obtained by endowing $$\mathbb{P}(\mathrm{w})$$ with a natural graded structure sheaf. The resulting graded ringed space $$\overline{\mathbb{P}}(\mathrm{w})$$ 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 $$\overline{\mathbb{P}}({\mathrm w})$$ a result which is very similar to Beilinson's theorem on $$\mathbb{P}^n$$, with the main difference that the resolution involves, besides $$\Omega_{\overline{\mathbb{P}}(\mathrm{w})}^j(j)$$ for $$0\le j\le n$$, also $$\mathcal{O}_{\overline{\mathbb{P}}(\mathrm{w})}(l)$$ for $$n-\sum_{i=0}^n\mathrm{w}_i< l< 0$$.

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 $$\mathbb{P}(\mathrm{w})$$, induced by $$4$$ sections $$\sigma_i\in H^0(S,\mathcal{O}_S(\mathrm{w}_iK_S))$$). 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 $$\overline{\mathbb{P}}(\mathrm{w})$$, 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 $$\mathbb{P}(1,1,2,3)$$.

• Beilinson's theorem on $$\bar{\mathbb{P}}(\textrm{w})$$
• Applications to surfaces with $$p_g=q=2$$, $$K^2=4$$