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 $3$â€“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 $\mathbb{P}(1,1,2,3)$.