The group comprises two units, each of which also carries out some independent research activity.

The group is intent on studying those problems in the theory of the spaces of moduli of the algebraic spin curves, or featuring other structures, which can be formulated also by means of the theory of congruences, with particular attention being paid to linear ones, to the Grassmannians of lines or, more generally, of the Fano subvarieties within those Grassmannians.

A case that will be dealt with is that of the quadratic complex X. The Hilbert space H of the lines of X is the Jacobian A of the genus 2 curve built by means of the double covering, achieved through the correspondence between the quadrics and the plane families they contain, of the quadric bundle having X as the base locus. The universal family U of H is a 4-to-1 cover of X, over a section of X given by a hypersurface of degree 8 of the projective space in which X is naturally immersed. The generic rational curve R of degree d contained in X is covered 4-to-1 by a curve C of genus 4d-3 contained in U. The natural projection of U over A causes a morphism of C over a noded curve M, whose nodes correspond to the bisecting lines of R. Also, the correspondence induced on C by the incidence relation of the lines of X meeting R builds a correspondence I over C that, in this case, is no theta correspondence, as in the analogous cases studied by Mukai, Takagi, Zucconi and others.

Indeed, an interesting problem is that of clarifying the nature of said correspondence I with C and of studying the rational application of the Hilbert scheme of the rational curves of degree d on X in the space of the moduli of pairs (C, I), where C is a tetragonal curve. This theory appears to be new, and in any case it is only a particular aspect of a broader theory. In fact, the quadratic complex X can be replaced by any Fano variety, through whose general point 4 distinct lines pass.

Another important example is the linear congruence obtained by intersecting the Grassmannian in IP^9 with four hyperplanes of IP^9. In this case, the Hilbert scheme of the lines of Y is a well-known variety: the Palatini threefold. By means of a systemic analysis of such situations, we expect to be able to build geometrically loci that are otherwise not easily visible, belonging to the moduli space of the quadrigonal curves. We deem it useful to highlight here that the success of our research, admittedly featuring very difficult stages and needing the integration of competences that are new for the group, too, and located beyond the national and European borders, will depend critically on the volume of funding the group will receive over the next few years.

We believe that, apart from the intrinsic beauty of it, the synthetic construction of some moduli spaces of algebraic curves (usually achieved by means of an analytical-algebraic approach), through geometrically elementary structures such as the rational curves in the Grassmann variety can, in due time, also turn out to be useful in order to solve problems of mechanical and effective calculus on said moduli spaces. This is commonly acknowledged as one of the fundamental problems in geometry.