In this work we study the moduli part in the canonical bundle formula of an lc-trivial fibration whose general fibre is a rational curve. If $r$ is the Cartier index of the fibre, it was expected that $12r$ would provide a bound on the denominators of the moduli part. Here we prove that such a bound cannot even be polynomial in $r$, we provide a bound $N\left(r\right)$ and an example where the smallest integer that clears the denominators of the moduli part is $N\left(r\right)/r$. Moreover we prove that even locally the denominators depend...

We take up the study of the Brill-Noether loci $W^r(L,X):=\{\eta \in {\mathrm{Pic}}^0\left(X\right)|h^0(L\otimes \eta )\ge r+1\}$, where $X$ is a smooth projective variety of dimension $>1$, $L\in \mathrm{Pic}\left(X\right)$, and $r\ge 0$ is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for $h^0\left(K_D\right)$, where $D$ is a divisor that moves linearly on a smooth projective variety $X$ of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension $>2$. In the $2$-dimensional case we prove an...

We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c 1 = 0, c 1...