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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$60
Individual Members: US$36
Institutional Members: US$48
Order Code: MEMO/183/862
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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 \(\mathbb{P}(\mathrm{w})\) (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 \(\mathbb{P}(1,1,2,3)\).

Table of Contents

  • Introduction
  • Graded schemes
  • Beilinson's theorem on \(\bar{\mathbb{P}}(\textrm{w})\)
  • The theorem on weighted canonical projections
  • Applications to surfaces with \(p_g=q=2\), \(K^2=4\)
  • Abelian categories and derived categories
  • Bibliography
  • Index
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