Related papers: On maximally inflected hyperbolic curves

We study the reciprocal position of nine points in the plane, according to their collinearities. In particular, we consider the case in which the nine points are contained in an irreducible cubic curve and we give their classification. If…

Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^2=-2$. We prove a higher-dimensional generalization conjectured by Hassett and…

We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal…

The fine curve graph of a surface is a graph whose vertices are essential simple closed curves and whose edges connect disjoint curves. Following a rich history of hyperbolicity of various graphs associated to surfaces, the fine curve graph…

We establish the equidistribution with respect to the bifurcation measure of post-critically finite maps in any one-dimensional algebraic family of unicritical polynomials. Using this equidistribution result, together with a combinatorial…

A projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. This is a property which follows from Kobayashi hyperbolicity. We prove…

We develop a reduction theory for the representation of $\mathrm{SL}_n$ on pairs of symmetric $n\times n$ matrices. We apply this theory to the pencils of quadrics arising from divisors on hyperelliptic curves. We use these results to show…

In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of $\mathbb{R}^{n+1}$…

Local diffusion of strictly hyperbolic higher-order PDE's with constant coefficients at all simple singularities of corresponding wavefronts can be explained and recognized by only two local geometrical features of these wavefronts. We…

Four points ordered in the positive order on the unit circle determine the vertices of a quadrilateral, which is considered either as a euclidean or as a hyperbolic quadrilateral depending on whether the lines connecting the vertices are…

We prove that for any pair of integers 0\leq r\leq g such that g\geq 3 or r>0, there exists a (hyper)elliptic curve C over F_2 of genus g and 2-rank r whose automorphism group consists of only identity and the (hyper)elliptic involution. As…

In this paper we obtain an explicit formula for the number of degree d curves in two dimensional complex projective space, passing through (d(d+3)/2 -k) generic points and having a codimension k singularity, where k is at most 7. In the…

This paper deals with singularities of genus 2 curves on a general (d_1,d_2)-polarized abelian surface (S,L). In analogy with Chen's results concerning rational curves on K3 surfaces [Ch1,Ch2], it is natural to ask whether all such curves…

A smooth curve in the real projective plane is hyperbolic if its ovals are maximally nested. By the Helton-Vinnikov Theorem, any such curve admits a definite symmetric determinantal representation. We use polynomial homotopy continuation to…

We study hyperbolicity for quasi-projective varieties where the boundary divisor consists of n+1 numerically parallel effective divisors on a complex projective variety of dimension n, allowing non-empty intersection. Under explicit local…

The restrictions on the topology of nonsingular plane projective real algebraic curves of odd degree, obtained by O. Viro and the author in the paper published in the early 90s, are extended to flexible curves lying on an almost complex…

In this paper we resolve the degree-2 Abel map for nodal curves. Our results are based on a previous work of the authors reducing the problem of the resolution of the Abel map to a combinatorial problem via tropical geometry. As an…

We deal with the distributions of holomorphic curves and integral points off divisors. We will simultaneouly prove an optimal dimension estimate from above of a subvariety W off a divisor D which contains a Zariski dense entire holomorphic…

We determine the versal deformation of cones, in the simplest case: cones over hyperelliptic curves of high degree. In particular, we show that for degree $4g+4$, the highest degree for which interesting deformations exist, the number of…

In this thesis, I am going to consider Bernoulli percolation on graphs admitting vertex-transitive actions of groups of isometries of d-dimensional hyperbolic spaces H^d. In the first chapter, I give an overview concerning percolation and…