FOSICAV · Families of Subvarieties in Complex Algebraic Varieties

In relation with the study of both moduli and enumerative problems incomplex algebraic geometry, we propose the geometric study of various families of subvarieties ofcertain complex algebraic varieties of small dimension, and mainly offamilies of (possibly singular) curves. The Severi varieties are atypical example: they parametrize curves of given degree and geometricgenus in the projective plane; the general such curve has a prescribednumber of ordinary double points and no further singularity. Apart from exploring their dimensions, smoothness, and irreducibilityproperties, we have in mind to determine their Hilbert polynomials (which among other things encode their degrees, the latter beingimportant enumerative invariants).A central feature of our project is to conduct this analysis bydegeneration: to study families of subvarieties in a given variety X,we let X degenerate and look at what happens in the limit. Forinstance, to study curves on a general K3 surface, we can let itdegenerate to a union of projective planes, the dual graph of which isa triangulation of the real 2-sphere.We shall consider the following kind of families of subvarieties:families of curves with prescribed invariants and singularities insurfaces (with special attention to the two cases of the projective plane,and of K3 surfaces), families of hyperplane sections with prescribedsingularities of hypersurfaces in projective spaces, families ofcurves with a given genus in Calabi-Yau threefolds, and families ofsurfaces in the projective 3-space containing curves with unexpectedsingularities.