Polarized minimal families of rational curves and higher Fano manifolds

In this paper we investigate Fano manifolds $X$ whose Chern characters $ch_k(X)$ satisfy some positivity conditions. Our approach is via the study of polarized minimal families of rational curves $(H_x,L_x)$ through a general point $x\in X$. First we translate positivity properties of the Chern characters of $X$ into properties of the pair $(H_x,L_x)$. This allows us to classify polarized minimal families of rational curves associated to Fano manifolds $X$ satisfying $ch_2(X)\geq0$ and $ch_3(X)\geq0$. As a first application, we provide sufficient conditions for these manifolds to be covered by subvarieties isomorphic to $\mathbb P^2$ and $\mathbb P^3$. Moreover, this classification enables us to find new examples of Fano manifolds satisfying $ch_2(X)\geq0$.Comment: 17 page

A modular description of X0(n)\mathscr{X}_0(n)

As we explain, when a positive integer $n$ is not squarefree, even over $\mathbb{C}$ the moduli stack that parametrizes generalized elliptic curves equipped with an ample cyclic subgroup of order $n$ does not agree at the cusps with the $\Gamma_0(n)$-level modular stack $\mathscr{X}_0(n)$ defined by Deligne and Rapoport via normalization. Following a suggestion of Deligne, we present a refined moduli stack of ample cyclic subgroups of order $n$ that does recover $\mathscr{X}_0(n)$ over $\mathbb{Z}$ for all $n$. The resulting modular description enables us to extend the regularity theorem of Katz and Mazur: $\mathscr{X}_0(n)$ is also regular at the cusps. We also prove such regularity for $\mathscr{X}_1(n)$ and several other modular stacks, some of which have been treated by Conrad by a different method. For the proofs we introduce a tower of compactifications $\overline{Ell}_m$ of the stack $Ell$ that parametrizes elliptic curves---the ability to vary $m$ in the tower permits robust reductions of the analysis of Drinfeld level structures on generalized elliptic curves to elliptic curve cases via congruences.Comment: 67 pages; final version, to appear in Algebra and Number Theor

Uniform families of minimal rational curves on Fano manifolds

It is a well-known fact that families of minimal rational curves on rational homogeneous manifolds of Picard number one are uniform, in the sense that the tangent bundle to the manifold has the same splitting type on each curve of the family. In this note we prove that certain --stronger-- uniformity conditions on a family of minimal rational curves on a Fano manifold of Picard number one allow to prove that the manifold is homogeneous

Moduli of sheaves: a modern primer

We give a modern introduction to the moduli of sheaves. After reviewing the classical theory, we give a catalogue of results from the last decade. We then consider a more "symmetric" formulation of the theory by working with gerbes from the start.Comment: 26 pages, contribution to proceedings of 2015 AMS Summer Institute in Algebraic Geometr