Let $\tilde{X}$ be a desingularization of a normal surface $X$. The group Pic$\left(\tilde{X}\right)$ is provided with an order relation $L\underset{\_}{\gg }0$, defined by $L$. $V\le 0$ for any effective exceptional divisor $V$. Comparing to the usual order relation we define the ceiling of $L$ which is an exceptional divisor. This notion allows us to improve the usual vanishing theorem and we deduce from it a numerical criterion for rationality and a genus formula for a curve on a normal surface; the difficulty lies in the case of a Weil divisor which is not a Cartier...

In this paper we explore several concrete problems, all more or less related to the intersection theory of the moduli space of (stable) curves, introduced by Mumford [Mu 1].

In this article, we complete the interpretation of groups of classes of invariant divisors on a complex toric variety X of dimension n in terms of suitable (co-) homology groups. In [BBFK], we proved the following result (see Satz 1 below): Let $ClDi{v}_C^{}\left(X\right)$ and $ClDi{v}_W^{}\left(X\right)$ denote the groups of classes of invariant Cartier resp. Weil divisors on X. If X is non degenerate (i.e., not equivariantly isomorphic to the product of a toric variety and a torus of positive dimension), then the natural homomorphisms $ClDi{v}_C^{}\left(X\right)\to H^2\left(X\right)$ and $ClDi{v}_W^{}\left(X\right)\to H_{2n-2}^{cld}\left(X\right)$ are...