" Algebraic subgroups of the Cremona group of the plane."

Abstract: The Cremona group of the plane is its group of birational transformations. Its connected algebraic subgroups have been classified by F. Enriques in 1893, they correspond to the automorphisms of minimal rational surfaces. I will describe this relation, and show how it extends to algebraic groups in general (not necessarily connected), and in particular to finite subgroups. The notion of algebraic subgroups is also related to family of transformations of the plane, which endows the Cremona group with a natural Zariski topology. The Cremona group does not contain any non-trivial closed normal subgroup, so it is simple endowed with its topology, but it is not simple as an abstract group." .

"Positive submanifolds of polarized manifolds."

Abstract: Let $(X,A)$ be a smooth complex polarized variety of dimension $k\geq 3$, and let $Y$ be a smooth (connected) subvariety of dimension $y\geq 1$. Then one can define the Seshadri constant $\varepsilon(Y,A)$ of $Y$ with respect to the polarization $A$ as $$\varepsilon(Y,A):=\sup\{\eta\in {\mathbb{Q}} \;|\; A^*-\eta E\;\;{\rm is\;ample}\},$$ where $A^*$ is the pull-back of $A$ on the blowing-up $X_Y$, $\pi:X_Y\to X$, of $X$ along $Y$ and $E$ is the exceptional divisor. Since $A$ is a polarization on $X$ and the line bundle $\mathscr O_{X_Y}(-E)$ is $\pi$-ample, this definition makes sense and yields the inequality $\varepsilon(Y,A)>0$. Then $\varepsilon(Y,A)$ is a strictly positive real number. If $k=3$ and $y=1$ this number was considered for good reasons by Paoletti [2] to study the so-called Seshadri positive curves in a polarized threefold. Subsequently, this theory was generalized and completed to the case when $Y$ is a curve in a smooth polarized variety $(X,A)$ of arbitrary dimension $k\geq 3$ in [1]. Specifically, for every $\eta\in(0,\varepsilon(Y,A))$ define the numerical invariant $$\delta_{\eta}(Y,A):=\eta^{k-3}\big(\eta\deg(N)-(k-2)d\big),$$ where $N$ is the normal bundle of $Y$ in $X$ and $d$ is the degree of $Y$ with respect to the polarization $A$. In this case $Y$ is said to be {\em Seshadri big} in $(X,A)$ if there is an $\eta\in(0,\varepsilon(Y,A))$ such that $\delta_{\eta}(Y,A)>0$. This definition has a natural geometrical interpretation. Moreover, at the end of [1], a possible generalization of Seshadri positivity for submanifods $Y$ of dimension $y\geq 2$ in a complex polarized manifold of dimension $k\geq 4$ has also been suggested. Unfortunately, that definition turned out to be much too strong to work with. Our aim is to give (in our opinion) the natural generalization of the concept of Seshadri positivity to submanifolds of dimension $y\geq 2$ in a smooth polarized manifold $(X,A)$. This allows us not only to construct many examples of Seshadri positive submanifolds of $(X,A)$, but also to generalize in arbitrary dimension all the results proved in [1], and to give further motivations for the study of Seshadri positivity. (Joint work with L. Badescu.)

[1] L. Badescu, M.C. Beltrametti and P. Francia, Positive curves in polarized manifolds, Manuscripta Math. 92 (1997), 369-388.

[2] R. Paoletti, Seshadri positive curves in a smooth projective 3-fold, Atti Accad. Naz. Lincei Rend., Matematica e Applicazioni, Ser. IX, vol. VI, (1995), 259-274.

"Some birational geometry of the Prym moduli spaces"

Abstract: A brief survey introducing the birational geometry of the Prym moduli spaces R_g is presented, with special regard to rational parametrizations. The survey is followed by the description of some new results in low genus g = 6, 7, 8 and their relations to the study of hyperplane sections of Nikulin surfaces.