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 (0le jle 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 (0le jle n), also (mathcal{O}_{overline{mathbb{P}}(mathrm{w})}(l)) for (n-sum_{i=0}^nmathrm{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_iin 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)).

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