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We develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves into surfaces defined by a polynomial equation: in particular,…
In the projective plane over a finite field of characteristic not equal to 2, we compute the probability that a randomly selected pair of distinct conics $(\mathscr{A},\mathscr{B})$, with $\mathscr{A}$ smooth or singular and $\mathscr{B}$…
We find sharp upper bounds on the order of the automorphism group of a hypersurface in complex projective space in every dimension and degree. In each case, we prove that the hypersurface realizing the upper bound is unique up to…
In this article, we construct countably many mutually non-isotopic diffeomorphisms of some closed non simply-connected 4-manifolds that are homotopic to but not isotopic to the identity, by surgery along $\Theta$-graphs. As corollaries of…
A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local-global principle for the…
We give a Belyi-type characterisation of smooth complete intersections of general type over $\mathbb{C}$ which can be defined over $\bar{\mathbb{Q}}$. Our proof uses the higher-dimensional analogue of the Shafarevich boundedness conjecture…
We settle the automorphism groups of curves appearing in a classification list of smooth plane curves with at least two Galois points. One of them is an ordinary curve whose automorphism group exceeds the Hurwitz bound.
In this work we are interested in the characterization of curves that belong to a given surface. To the best of our knowledge, there is no known general solution to this problem. Indeed, a solution is only available for a few examples:…
We consider quadric surface fibrations over curves, defined over algebraically closed and finite fields. Our goal is to understand, in geometric terms, spaces of sections for such fibrations. We analyze varieties of maximal isotropic…
The simplest patterns of qualitative changes on the configurations of lines of principal curvature} around umbilic points on surfaces whose immersions into $\mathbb R^3$ depend smoothly on a real parameter (codimension one umbilic…
We compute the automorphism scheme of a generic odd dimensional $(2,2)$-complete intersection in characteristic $2$. This is the only case for complete intersections having a non-trivial identity component in automorphism schemes apart from…
We study transformations as in the title with emphasis on those having smooth connected base locus, called "special". In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by…
We discuss the bifurcation structure of homoclinic orbits in bimodal one dimensional maps. The universal structure of these bifurcations with singular bifurcation points and the web of bifurcation lines through the parameter space are…
The Eckardt hypersurface in $\mathbb{P}^{19}$ parameterizes smooth cubic surfaces with an Eckardt point, which is a point common to three of the $27$ lines on a smooth cubic surface. We describe the cubic surfaces lying on the singular…
We study the arithmetic of bitangents of smooth quartics over global fields. With the aid of computer algebra systems and using Elsenhans--Jahnel's results on the inverse Galois problem for bitangents, we show that, over any global field of…
This research monograph focuses on the arithmetic, over number fields, of surfaces fibred into curves of genus 1 over the projective line, and of intersections of two quadrics in projective space. The first half takes up and develops…
We study the geometric and combinatorial effect of smoothing an intersection point in a collection of arcs or curves on a surface. We prove that all taut arcs with fixed endpoints and all taut 1-manifolds with at least two non-disjoint…
Orbifolds of two-dimensional quantum field theories have a natural formulation in terms of defects or domain walls. This perspective allows for a rich generalisation of the orbifolding procedure, which we study in detail for the case of…
In this paper, we prove some fundamental theorems for holomorphic curves on angular domain intersecting a hypersurface, finite set of fixed hyperplanes in general position and finite set of fixed hypersurfaces in general position on complex…
There is a natural bijection between Dyck paths and basis diagrams of the Temperley-Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the…