Related papers: Characterizing $(d,h)$-elliptic stable irreducible…
We study infinitesimal deformations of complete hyperbolic surfaces with boundary and with ideal vertices, possibly decorated with horoballs. ``Admissible'' deformations are the ones that pull all horoballs apart; they form a convex cone of…
We consider the problem of enumerating d-irreducible maps, i.e. planar maps whose all cycles have length at least d, and such that any cycle of length d is the boundary of a face of degree d. We develop two approaches in parallel: the…
We construct a well-behaved compactification of finite covers of a stacky curve using admissible cover degenerations. Using our construction, we compactify the space of tetragonal curves on Hirzebruch surfaces. As an application, we…
The moduli stack of Deligne-Mumford stable curves of genus g admits a stratification, so that the number of nodes of the curves belonging to one stratum is constant. The irreducible components of the stratum corresponding to curves with…
Denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, that is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb P^r$. A component…
We report on the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer $d$ and a finite…
We give effective upper bounds for the number of purely inseparable points on non isotrivial curves over function fields of positive characteristic and of transcendence degree one. These bounds depend on the genus of the curve, the genus of…
This article deals with the Galois representation attached to elliptic curves with an isogeny of prime degree over a number field. We first determine uniform criteria for the irreducibility of Galois representations attached to elliptic…
Given a smooth cubic hypersurface $X$ over a finite field of characteristic greater than 3 and two generic points on $X$, we use a function field analogue of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number…
We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.
We characterize contractible curves on proper normal algebraic surfaces in terms of complementary Weil divisors. Using this we generalize the classical criteria of Castelnuovo and Artin. As application we derive a finiteness result on…
We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the…
Let $f: X \to Y$ be a regular covering of a surface $Y$ of finite type with nonempty boundary, with finitely-generated (possibly infinite) deck group $G$. We give necessary and sufficient conditions for an integral homology class on $X$ to…
The `linear orbit' of a plane curve of degree d is its orbit in P^{d(d+3)/2} under the natural action of PGL(3). We classify curves with positive dimensional stabilizer, and we compute the degree of the closure of the linear orbits of…
In the last decades there have been introduced different concepts of stability for projective varieties. In this paper we give a natural and intrinsic criterion of the Chow, and Hilbert, stability for complex irreducible smooth projective…
We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain…
We construct a good compactification of the variety of irreducible projective plane curves of degree n with d nodes and no other singularities.
The goal of this paper is to motivate a boundedness conjecture on nearby slopes of $\ell$-adic sheaves in positive characteristic, and to prove it for smooth curves. For a constructible $\ell$-adic sheaf, we prove the finiteness of the set…
Let $\mathcal{H}_{d,g,r}$ be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^r.$ We denote by $\mathcal{H}^\mathcal{L}_{d,g,r}$ the union of those components of…
We compute the stable cohomology of moduli spaces of hyperelliptic curves of a fixed genus embedded on a fixed Hirzebruch surface. We also describe these moduli spaces of embedded hyperelliptic curves in terms of moduli spaces of pointed…