Related papers: On curves lying on a rational normal surface scrol…

We show that a rational normal scroll can in general be set-theoretically defined by a proper subset of the 2-minors of the associated two-row matrix. This allows us to find a class of rational normal scrolls that are almost set-theoretic…

Let X be a zero-dimensional scheme in P1 \times P1. Then X has a minimal free resolution of length 2 if and only if X is ACM. In this paper we determine a class of reduced schemes whose resolutions, similarly to the ACM case, can be…

The paper is a generalization of a result of I. Dolgachev, M. Mendes Lopes, and R. Pardini. We prove that a smooth projective complex surface $X$, not necessarily minimal, contains $h^{1,1}(X)-1$ disjoint $(-2)$-curves if and only if $X$ is…

A well-known conjecture asserts that smooth threefolds $X\subset\{\mathbb P}^5$ are quadratically normal with the only exception of the Palatini scroll. As a corollary of a more general statement we obtain the following result, which is…

We give a short proof of the following result: Let $X$ be a complex surface of general type. If the canonical divisor of the minimal model of $X$ has selfintersection $= 1$, then $X$ is not diffeomorphic to a rational surface. Our proof is…

For $q\leq 3$ smooth plane algebraic curves $\mathcal{C}_i$ having simple normal crossings, if the invariant logarithmic $2$-jet differential bundle associated to $(\mathbb{P}^2(\mathbb{C}), \sum_{i=1}^q \mathcal{C}_i)$ has a nonzero…

The purpose of this article is to study the deformations of smooth surfaces $X$ of general type whose canonical map is a finite, degree 2 morphism onto a minimal rational surface or onto $\mathbf F_1$, embedded in projective space by a very…

We study the geometry and arithmetic of so-called primary Burniat surfaces, a family of surfaces of general type arising as smooth bidouble covers of a del Pezzo surface of degree 6 and at the same time as \'etale quotients of certain…

Let $S$ be a smooth projective surface on a smooth threefold $X$ such that $X$ has Picard rank 1 and NS$(S)$ is generated by the restriction of divisors from X. We show that if $X$ satisfies the Bogomolov-Gieseker type inequality for tilt…

Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface and $G$ be a finite subgroup in $\operatorname{Aut}(X)$. In this paper we study when the quotient surface $X / G$ can be non-rational over $\Bbbk$. Obviously, if…

Let $X$ be a minimal surface of general type over an algebraically closed field $\mathbf{k}$ of $\mathrm{char}.(\mathbf{k})=p\ge 0$. If the Albanese morphism $a_X:X\to \mathrm{Alb}_X$ is generically finite onto its image, we formulate a…

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe…

We extend Reider's freeness criterion to normal surfaces of characteristic 0. Let Y be a normal surface. Let D be a nef divisor on Y such that K_Y+D is a Cartier divisor. Let x be a point on Y. If x is a base point of |K_Y+D| and…

We determine the central simple algebras D over a functionfield K of trancendence degree two which admit a model of smooth Cayley-Hamilton algebras. This happens if and only if there is a smooth model S of K such that the ramification…

A recent result of Eisenbud-Schreyer and Boij-S\"oderberg proves that the Betti diagram of any graded module decomposes as a positive rational linear combination of pure diagrams. When does this numerical decomposition correspond to an…

We study the arithmetic self-intersection number of the dualizing sheaf on arithmetic surfaces with respect to morphisms of a particular kind. We obtain upper bounds for the arithmetic self-intersection number of the dualizing sheaf on…

We study the moduli space of stable sheaves of Euler characteristic 1, supported on curves of arithmetic genus 2 contained in a smooth quadric surface. We show that this moduli space is rational. We give a classification of the stable…

ACM rank 1 bundles on del Pezzo surfaces are classified in terms of the rational normal curves that they contain. A complete list of ACM line bundles is provided. Moreover, for any del Pezzo surface $X$ of degree less or equal than six and…

We provide two examples of smooth projective surfaces of tame CM type, by showing that any parameter space of isomorphism classes of indecomposable ACM bundles with fixed rank and determinant on a rational quartic scroll in projective…

We study the Betti tables of reducible algebraic curves with a focus on connected line arrangements and provide a general formula for computing the quadratic strand of the Betti table for line arrangements that satisfy certain hypotheses.…