Abstract:
Given a smooth projective curve X, we give effective very ampleness bounds for generalized theta divisors on the moduli spaces $SU_X(r,d)$ and $U_X(r,d)$ of semistable vector bundles of rank r and degree d on X with fixed, respectively arbitrary, determinant.

Abstract:
Let X be a projective variety which is algebraic Lang hyperbolic. We show that Lang's conjecture holds (one direction only): X and all its subvarieties are of general type and the canonical divisor K_X is ample at smooth points and Kawamata log terminal points of X, provided that K_X is Q-Cartier, no Calabi-Yau variety is algebraic Lang hyperbolic and a weak abundance conjecture holds.

Abstract:
The twisted homogeneous coordinate ring is one of the basic constructions of the noncommutative projective geometry of Artin, Van den Bergh, and others. Chan generalized this construction to the multi-homogeneous case, using a concept of right ampleness for a finite collection of invertible sheaves and automorphisms of a projective scheme. From this he derives that certain multi-homogeneous rings, such as tensor products of twisted homogeneous coordinate rings, are right noetherian. We show that right and left ampleness are equivalent and that there is a simple criterion for such ampleness. Thus we find under natural hypotheses that multi-homogeneous coordinate rings are noetherian and have integer GK-dimension.

Abstract:
Generalizing work done by Miyaoka and others in the case of divisors and of curves, we compute the cones of effective cycles of arbitrary dimension on a projective bundle over a complex projective curve in terms of the numerical data in an associated Harder-Narasimhan filtration. An application to cycles on projective bundles over a smooth complex projective base of arbitrary dimension is also given.

Abstract:
We systematically study the splitting of vector bundles on a smooth, projective variety, whose restriction to the zero locus of a regular section of an ample vector bundle splits. First, we find ampleness and genericity conditions which ensure that the splitting of the vector bundle along the subvariety implies its global splitting. Second, we obtain a simple splitting criterion for vector bundles on the Grassmannian and on partial flag varieties.

Abstract:
We investigate which of the characterizations of integral ample divisors can be extended to $\mathbb{Q}$-divisors and $\mathbb{R}$-divisors when associating to the Weil divisor the line bundle of its integral part.

Abstract:
Very ampleness criteria for rank 2 vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.

Abstract:
Taking inspiration from a recent breakthrough of Damian Brotbek, for the intersection family $\mathcal{X}$ of certain Fermat-type hypersurfaces in $\mathbb{P}_{\mathbb{K}}^N$ defined over any field $\mathbb K$, we construct explicit symmetric differential forms by applying Cramer's rule, skipping cohomology arguments, and we further exhibit unveiled families of lower degree symmetric differential forms on all possible intersections of $\mathcal{X}$ with coordinate hyperplanes. Thereafter, we develop what we call the moving coefficients method to prove a conjecture made by Olivier Debarre: for a generic choice of $c\geqslant N/2$ hypersurfaces $H_1,\dots,H_c\subset \mathbb{P}_{\mathbb K}^N$ of degrees $d_1,\dots,d_c$ sufficiently large, the intersection $X:=H_1 \cap \cdots \cap H_c $ has ample cotangent bundle $\Omega_X$, and concerning effectiveness, the lower bound: \[ d_1,\dots,d_c\,\geqslant\,N^{N^2} \] works. In particular, when $\mathbb{K}=\mathbb{C}$, it is known that such $X$ are Kobayashi-hyperbolic. Lastly, assuming that the ambient field $\mathbb{K}$ has characteristic zero, thanks to known results about the Fujita Conjecture, we establish the very-ampleness of $\mathsf{Sym}^{\kappa}\,\Omega_X$ for all $\kappa\geqslant \kappa_0$, with a uniform lower bound: \[ \kappa_0\, =\, 16\, (N-c+1)^2\, \Big( \sum_{i=1}^c\, d_i \Big)^2 \] independent of $X$.

Abstract:
Let $X$ be a compact quotient of a bounded domain in $\mathbb C^n$. Let $K_X$ be the canonical line bundle of $X$. In this paper, we shall introduce the notion of $S$ very ampleness for the pluri-canonical line bundles $mK_X$ by using the Poincar\'e series. The main result is an effective Seshadri constant criterion of $S$ very ampleness for $mK_X$. An elementary proof of surjectivity of the Poincar\'e map is also given.