Relative Quot schemes over families of smooth and nodal curves

The present work studies relative Quot schemes parameterizing locally free sheaf quotients with zero--dimensional support on the fibers of a morphism. We focus on families of smooth curves with at most nodal singularities, as well as families of smooth, higher dimensional varieties. The thesis is organized around three projects, which explore both geometric and combinatorial aspects of these relative Quot schemes. The first project, carried out in collaboration with Barbara Fantechi and Ajay Gautam, provides an explicit geometric description of Quot schemes of families of smooth projective curves. We show that the Quot scheme of a locally free sheaf filtered by line bundles admits a finite locally closed stratification, each stratum being isomorphic to an affine vector bundle over products of symmetric powers of the family, with rank given by an explicit formula. Our result generalizes an existing description by Bifet via an independent approach, which does not rely on Białynicki--Birula decompositions. In addition, it determines the class of the relative Quot scheme in the Grothendieck ring of varieties. Further applications concern nested Quot schemes and Quot schemes of positive rank quotients on a smooth projective curve. The second project addresses Quot schemes of smooth morphisms of arbitrary relative dimension. By analyzing the Quot--to--Sym morphism and its behavior under the natural stratification by integer partitions, we show that its restriction to each stratum is {\'e}tale–locally trivial. Combined with the language of power structures on the Grothendieck ring, this result yields two formulas for Quot schemes of smooth morphisms. The third project concerns Quot schemes over families of nodal curves, with emphasis on Losev--Manin spaces. We first study the geometry of Hilbert schemes over Losev--Manin spaces, showing that they are smooth, irreducible of known dimension. Moreover, by combining existing formulas for smooth curves and nodal singularities, we derive an explicit identity for generating functions of the corresponding Quot scheme classes in the Grothendieck ring. The chapter concludes by addressing some open questions and possible further research directions.