Related papers: A note on the linear systems on the projective bun…

In this book, we establish a theory of adelic line bundles over quasi-projective varieties over finitely generated fields. Besides definitions of adelic line bundles, we consider their intersection theory, volume theory, and height theory,…

While higher bundles are of clear relevance to higher gauge theory, examples other than abelian bundle gerbes are hard to come across. One would in particular like to see 2-bundles where the structure 2-group is the String 2-group…

The abelian sandpile models feature a finite abelian group $G$ generated by the operators corresponding to particle addition at various sites. We study the canonical decomposition of $G$ as a product of cyclic groups $G = Z_{d_1} \times…

We reduce the problem of the projective normality of polarized abelian varieties to check the rank of very explicit matrices. This allow us to prove some results on normal generation of primitive line bundles on abelian threefolds and…

We give effective bounds on the generation of pushforwards of log-pluricanonical bundles twisted by ample line bundles. This gives a partial answer to a conjecture proposed by Popa and Schnell. We prove two types of statements: first, more…

I give a necessary and sufficient condition for a nef and big line bundle in positive characteristic to be semi-ample, and then give two applications: I show that the relative dualizing sheaf of the universal curve is semi-ample, in…

We classify principal bundles over anti-affine schemes with affine and commutative structural group. We show that this yields the classification of quasi-abelian varieties over a field k (i.e., group k-schemes with no non constant global…

Let C be a smooth projective curve of genus at least 2 over a field k. Given a line bundle L on C, we consider the moduli stack of rank 2n vector bundles E on C endowed with a nowhere degenerate symplectic form $b: E \otimes E \to L$ up to…

We prove some results on effective very ampleness and projective normality for some varieties with trivial canonical bundle. In the first part we prove an effective projective normality result for an ample line bundle on regular smooth…

Let X be a smooth projective complex variety, and L a line bundle on X . We say that the linear system |L| has maximal variation if its elements have the maximum number dim|L| of moduli. We discuss some cases where this situation is…

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety, $a\colon X\rightarrow A$ a morphism to an abelian variety such that $\rm{Pic}^0(A)$ injects into $\rm{Pic}^0(T)$ and let $L$ be a line bundle on $X$. Denote by…

Let $A$ be an abelian surface over an algebraically closed field $\overline{k}$ with an embedding $\overline{k}\hookrightarrow\mathbb{C}$. When $A$ is isogenous to a product of elliptic curves, we describe a large collection of pairwise…

The bad locus of a base-point free linear system L on a normal complex projective variety X is defined as the subset B(L) of X of points that are not contained in any irreducible and reduced member of L. In this paper we provide a geometric…

We study smooth projective complex varieties with ample cotangent bundle. Our main result is that in an abelian variety of dimension n, a complete intersection of at least n/2 general hypersurfaces of sufficiently high degrees has ample…

Let $X$ be a smooth projective $n$-fold such that $q(X)=0$ and $L$ a globally generated, big line bundle on $X$ such that $h^0(K_X+(n-2)L) >0$. We give necessary and sufficient conditions for the adjoint systems $|K_X+kL|$ to be birational…

Let $X$ be a smooth projective variety of dimension $5$ and $L$ be an ample line bundle on $X$. We show that $|K_X + 6L|$ is base-point free.

Let $X$ be a smooth projective variety of dimension $n\geq 3$, and let $L$ be an ample line bundle on $X$. In this article, we study the algebraic hyperbolicity of a very general section of the adjoint linear series $|K_X+mL|$ when the…

In this paper we study sum-free sets of order $m$ in finite Abelian groups. We prove a general theorem on 3-uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting.…

Let $X/\mathbb{F}_{q}$ be a smooth geometrically connected variety. Inspired by work of Corlette-Simpson over $\mathbb{C}$, we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on $X$ come from…

Let $A$ be an abelian scheme of dimension at least four over a $\mathbb{Z}$-finitely generated integral domain $R$ of characteristic zero, and let $L$ be an ample line bundle on $A$. We prove that the set of smooth hypersurfaces $D$ in $A$…