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In this paper, we study plane quintic curves whose automorphism groups have order greater than 10, as well as those with cyclic automorphism groups of order 8 and 10. The latter two cases are represented as one-parameter families, where…
We consider a ranking and selection (R&S) problem with the goal to select a system with the largest or smallest expected performance measure among a number of simulated systems with a pre-specified probability of correct selection. Fully…
For each proper divisor d of (r^2-r+1), r being a power of a prime, maximal curves over a finite field with r^2 elements covered by the Hermitian curve of genus 1/2((r^2-r+1)/d-1) are constructed.
We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. In doing so, we make fundamental connections with both discrete geometry and algebraic combinatorics. In…
A finite linear space is a finite set of points and lines, where any two points lie on a unique line. Well known examples include projective planes. This project focuses on linear spaces which admit certain types of symmetries. Symmetries…
We give a full classification of higher order parallel surfaces in three-dimensional homogeneous spaces with four-dimensional isometry group, i.e. in the so-called Bianchi-Cartan-Vranceanu family. This gives a positive answer to a…
The girth of a finitely generated group G is the supremum of the girth of Cayley graphs for G over all finite generating sets. Let G be a finitely generated subgroup of the mapping class group Mod(S), where S is a compact orientable…
We develop a constructive process which determines all extreme points of the unit ball of the space of $m$--linear forms, $m\geq1.$ Our method provides a full characterization of the geometry of that space through finitely many elementary…
In this paper, we answer a question posed by Kurt Johansson, to find a PDE for the joint distribution of the Airy Process. The latter is a continuous stationary process, describing the motion of the outermost particle of the Dyson Brownian…
We show that a general canonical curve is uniquely determined by the finite set of hyperplanes cutting theta-characteristics on it. Geometrical and combinatorial properties of the moduli space of stable spin curves are proved, which play an…
Using the fact that the Airy process describes the limiting fluctuations of the Hammersley last-passage percolation model, we prove that it behaves locally like a Brownian motion. Our method is quite straightforward, and it is based on a…
For a complex quasi-projective manifold with a finite group action, we define higher order generalized Euler characteristics with values in the Grothendieck ring of complex quasi-projective varieties extended by the rational powers of the…
We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed, or fully minimal. The type of $A$ depends on the normalized Weil numbers of $A$ and its…
We investigate the Jacobian decomposition of some algebraic curves over finite fields with genus $4$, $5$ and $10$. As a corollary, explicit equations for curves that are either maximal or minimal over the finite field with $p^2$ elements…
In this notes we study complex projective plane curves whose graded module of Jacobian syzygies is generated by its minimal degree component. Examples of such curves include the smooth curves as well as the maximal Tjurina curves. However,…
A plane curve is called strange if its tangent line at any smooth point passes through a fixed point, called the strange point. In this paper, we study $\mathbb{A}^1$-curves on the complement of a rational strange curve of degree $p$ in…
A ribbon is a first-order thickening of a non-singular curve. Motivated by a question of Eisenbud and Green, we show that a compactification of the moduli space of line bundles on a ribbon is given by the moduli space of semi-stable…
In a recent paper arXiv:1602.02300v2, Cook II, Harbourne, Migliore and Nagel related the splitting type of a line arrangement in the projective plane to the number of conditions imposed by a general fat point of multiplicity $j$ to the…
An abelian variety admits only a finite number of isomorphism classes of principal polarizations. The paper gives an interpretation of this number in terms of class numbers of definite Hermitian forms in the case of a product of elliptic…
We consider families of smooth projective curves of genus 2 with a single point removed and study their integral points. We show that in many such families there is a dense set of fibres for which the integral points can be effectively…