# The Beilinson Complex and Canonical Rings of Irregular Surfaces

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*Alberto Canonaco*

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

# Table of Contents

## The Beilinson Complex and Canonical Rings of Irregular Surfaces

- Contents v6 free
- Introduction 110 free
- Chapter 1. Graded schemes 615 free
- Chapter 2. Beilinson's theorem on P(w) 2938
- 2.1. Koszul complex and sheaves of differentials 3140
- 2.2. The theorem as equivalence of categories 3342
- 2.3. Morphisms between sheaves of differentials 3645
- 2.4. Uniqueness of the minimal resolution 3948
- 2.5. Explicit form of the minimal resolution 4150
- 2.6. Some computations of the minimal resolution 4756
- 2.7. The theorem on P(w) 5059
- 2.8. Application: a splitting criterion 5160

- Chapter 3. The theorem on weighted canonical projections 5362
- Chapter 4. Applications to surfaces with P[sub(g)] = q = 2, K[sup(2)] = 4 7079
- Appendix A. Abelian categories and derived categories 8796
- Bibliography 95104
- Index 97106 free