Related papers: Negative curves on algebraic surfaces

In this paper we give an upper bound on the number of rational points on an irreducible curve $C$ of degree $\delta$ defined over a finite field $\mathbb{F}_q$ lying on a Frobenius classical surface $S$ embedded in $\mathbb{P}^3$. This…

A classical example of Mumford gives a generically non-reduced component of the Hilbert scheme of smooth curves in the projective 3-space such that a general element of the component is contained in a smooth cubic hypersurface in the…

We prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_q$ with $d\leq q+1$, then there is an $\mathbb{F}_q$-line $L$ that intersects $C$ transversely. We also prove the same result for…

We study rational cuspidal curves in projective surfaces. We specify two criteria obstructing possible configurations of singular points that may occur on such curves. One criterion generalizes the result of Fernandez de Bobadilla, Luengo,…

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…

In this paper we focus on dynamical properties of (real) convex projective surfaces. Our main theorem provides an asymptotic formula for the number of free homotopy classes with roughly the same renormalized Hilbert length for two distinct…

In earlier work, we constructed invariants of irreducible representations of the Kauffman skein algebra of a surface. We introduce here an inverse construction, which to a set of possible invariants associates an irreducible representation…

We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke…

We consider rational surfaces $Z$ defined by divisorial valuations $\nu$ of Hirzebruch surfaces. We introduce the concepts of non-positivity and negativity at infinity for these valuations and prove that these concepts admit nice local and…

Let $C$ be a compact complex curve included in a non-singular complex surface such that the normal bundle is topologically trivial. Ueda studied complex analytic properties of a neighborhood of $C$ when $C$ is non-singular or is a rational…

Let $\Sigma$ be a smooth projective surface, let $f' : S' \to \Sigma$ be a double cover of $\Sigma$ and let $\mu : S \to S'$ be the canonical resolution. Put $f = f'\circ\mu$. An irreducible curve $C$ on $\Sigma$ is said to be a splitting…

For a smooth quasi-projective surface S over complex numbers we consider the Borel-Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative algebra by a version of the Hall…

It has been conjectured that every algebraic curve may be defined either over its field of moduli or over an extension of degree two of it. In this paper we provide a negative answer to it by giving examples of hyperelliptic curves which…

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…

We give conceptual proofs of some results on the automorphism group of an Enriques surface X, for which only computational proofs have been available. Namely, there is an obvious upper bound on the image of Aut(X) in the isometry group of…

We compute explicit rational models for some Hilbert modular surfaces corresponding to square discriminants, by connecting them to moduli spaces of elliptic K3 surfaces. Since they parametrize decomposable principally polarized abelian…

I prove "Lefschetz principle"-type theorems for semistable and curve semistable Higgs sheaves on smooth projective varieties defined over an algebraically closed field of characteristic $0$. These theorems are applied to reduce a…

Rational algebraic curves have been intensively studied in the last decades, both from the theoretical and applied point of view. In applications (e.g. level curves, linear homotopy deformation, geometric constructions in computer aided…

Let $\mathcal C$ be a real plane algebraic curve defined by the resultant of two polynomials (resp. by the discriminant of a polynomial). Geometrically such a curve is the projection of the intersection of the surfaces $P(x,y,z)=Q(x,y,z)=0$…

Let $X$ be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on $X$ in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An…