The Beilinson Complex and Canonical Rings of Irregular Surfaces

An important theorem by Beilinson describes the bounded derived category of coherent sheaves on P, yielding in particular a resolution of every coherent sheaf on P in terms of the vector bundles Ω Pn (j) for 0 ≤ j ≤ n. This theorem is here extended to weighted projective spaces. To this purpose we consider, instead of the usual category of coherent sheaves on P(w) (the weighted projective space of weights w = (w0, . . . ,wn)), a suitable category of graded coherent sheaves (the two categories are equivalent if and only if w0 = · · · = wn = 1, i.e. P(w) = P), obtained by endowing P(w) with a natural graded structure sheaf. The resulting graded ringed space P(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 P(w) a result which is very similar to Beilinson’s theorem on P, with the main difference that the resolution involves, besides Ω P(w) (j) for 0 ≤ j ≤ n, also O P(w)(l) for n− ∑n i=0 wi < 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 P(w), induced by 4 sections σi ∈ H(S,OS(wiKS))). This is a generalization of a theorem by Catanese and Schreyer (who treated the case of projections into P3), 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 P(w), satisfying some suitable conditions. Such a morphism is then explicitly determined in chapter 4 for a family of surfaces with numerical invariants pg = q = 2, K2 = 4, projected into P(1, 1, 2, 3).