Related papers: A Bertini type theorem for pencils over finite fie…
We compute the Artin $L$-function of a diagonal hypersurface D_{\lambda} over a finite field associated to a character of a finite group acting on D_{\lambda} , and under some condition, express it in terms of hypergeometric functions and…
We make a systematic study of the focal surface of a congruence of lines in the projective space. Using differential techniques together with techniques from intersection theory, we reobtain in particular all the invariants of the focal…
We completely describe the Brill-Noether theory of pencils on general primitive covers of elliptic curves of any degree.
We propose a simple method for uniform sampling of points on the surface of a hypersphere in arbitrarily many dimensions. By avoiding the evaluation of computationally expensive functions like logarithms, sines, cosines, or higher order…
We use function field analytic number theory to establish the irreducibility and dimension of the moduli space that parameterises morphisms of fixed degree from $\mathbb{P}^2$ to an arbitrary smooth hypersurface of sufficiently small…
A result of Beauville states that with a few positive characterstic exceptions, the smooth hyperplane sections of hypersurfaces of degree $d>2$ in projective space are not all isomorphic. We address the question of whether these sections…
Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…
We consider overdetermined problems of Serrin's type in convex cones for (possibly) degenerate operators in the Euclidean space as well as for a suitable generalization to space forms. We prove rigidity results by showing that the existence…
We establish an effective version of Schmidt's subspace theorem on a smooth projective variety $\mathcal{X}$ over function fields of characteristic zero for hypersurfaces located in m-subgeneral position with respect to $\mathcal{X}$. Our…
We classify the solutions to an overdetermined elliptic problem in the plane in the finite connectivity case. This is achieved by establishing a one-to-one correspondence between the solutions to this problem and a certain type of minimal…
In this work we study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve C, infinite dihedral covers, and pencils of curves containing C.
We prove a new Bertini-type Theorem with explicit control of the genus, degree, height, and the field of definition of the constructed curve. As a consequence we provide a general strategy to reduce certain height and rank estimates on…
With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral…
In this paper, our purpose is to study rigidity theorems for $\lambda$-hypersurfaces in Euclidean space under Gauss map. As a Bernstein type problem for $\lambda$-hypersurfaces, we prove that an entirely graphic $\lambda$-hypersurface in…
We study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of…
As an application of P. Delgine's theorem (Esnault and Kerz in Acta Math. Vietnam. 37:531-562, 2012) on a finiteness of $l$-adic sheaves on a variety over a finite field, we show the finiteness of \'etale coverings of such a variety with…
These are notes on the theory of super Riemann surfaces and their moduli spaces, aiming to collect results that are useful for a better understanding of superstring perturbation theory in the RNS formalism.
We study pencils of plane cubics with only one base point and general member smooth, giving a complete classification. Under the additional hypothesis that all members are irreducible, we prove that there exists a unique non-isotrivial…
Assume that the section conjecture holds over number fields. We prove then that it holds for a broad class of curves defined over finitely generated extensions of $\mathbb{Q}$. This class contains every projective, hyperelliptic curve,…
In this paper, we explain a simple and uniform construction of a smooth integral model associated to a quadratic, (anti)-hermitian, and (anti)-quaternionic hermitian lattice defined over an arbitrary local field. As one major application,…