Related papers: The relative canonical resolution: Macaulay2-packa…

We consider coverings of real algebraic curves to real rational algebraic curves. We show the existence of such coverings having prescribed topological degree on the real locus. From those existence results we prove some results on…

For smooth projective 3-folds of general type, we prove that the relative canonical stability $\mu_s(3)\leq 8$. This is induced from our improved result of Koll\'ar: the m-canonical map of a smooth projective 3-fold of general type is…

We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum…

We show the existence of toric resolution tower for an irreducible curve singularity which is explicitly described by Tschirnhausen polynomials. We deduce for a smooth affine plane curve from its topology restrictions for its singularity at…

We obtain a complete description of the effective cone of $C_{g-2}$ when $C$ is a general curve of genus $g \geq 6,$ as well as a new bound in the case where $C$ is a smooth plane quintic. In addition, we obtain a new virtual bound for the…

We prove three results on pure resolutions of vector bundles on projective spaces. First, we show that there are simple vector bundles of rank n on Pn with arbitrary homological dimension. We then analyze the pure resolutions given by the…

For each squarefree monomial ideal $I\subset S = k[x_{1},\ldots, x_{n}] $, we associate a simple graph $G_I$ by using the first linear syzygies of $I$. In cases, where $G_I$ is a cycle or a tree, we show the following are equivalent: (a) $…

Let $X$ be a compact connected Riemann surface of genus $g$, with $g\geq 2$, and ${\cal M}_{\xi}$ a smooth moduli space of fixed determinant semistable vector bundles of rank $n$, with $n\geq 2$, over $X$. Take a smooth anticanonical…

We define a class of plane curves which are close to the free divisors and such that conjecturally it contains the class of rational cuspidal curves. Using a recent result by U. Walther we show that any unicuspidal rational curve with a…

Let $C \subset P^{g-1}$ be a smooth canonical curve of genus $g \geq 3$. The purpose of this article is to further develop a method to classify varieties having $C$ as their curve section, using Gaussian map computations. In a previous…

We consider the space of holomorphic maps from a compact Riemann surface to a projective space blown up at finitely many points. We show that the homology of this mapping space equals that of the space of continuous maps that intersect the…

We show the existence of metrically dense entire curves in rationally connected complex projective manifolds confirming for this case a conjecture according to which such entire curves on projective manifolds exist if and only if these are…

We define the dimension 2g-1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P^1 with given ramification over infinity and sufficiently many fixed ramification points…

In this paper, we give an elementary new method for determining the rational points on algebraic curves using torsion packets. We also provide examples of curves for which all rational points can be completely determined by our method.

A point $p$ on a smooth complex projective curve $C$ of genus $g>3$ is subcanonical if the divisor $(2g-2)p$ is canonical. In the moduli space of pointed curves, the subcanonical locus is described by pairs $(C,p)$ as above, and it consists…

We investigate compactified Deligne-Lusztig varieties whose canonical divisor, when expressed as a linear combination of boundary divisors, has all coefficients strictly negative or zero. In dimension two we obtain explicit descriptions:…

Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower…

Let SU_C(2) be the moduli space of rank 2 semistable vector bundles with trivial de terminant on a smooth complex algebraic curve C of genus g > 1, we assume C non-hyperellptic if g > 2. In this paper we construct large families of pointed…

Fix integers $a\geq 1$, $b$ and $c$. We prove that for certain projective varieties $V\subset{\bold P}^r$ (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing…

This paper is devoted to the investigation of selected situations when the computation of projective (and other) equivalences of algebraic varieties can be efficiently solved with the help of finding projective equivalences of finite sets…