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By the famous ADE classification rational double points are simple. Rational triple points are also simple. We conjecture that the simple normal surface singularities are exactly those rational singularities, whose resolution graph can be…

We investigate geometric invariants of cuspidal edges on focal surfaces of regular surface. In particular, we shall clarify the sign of the singular curvature at a cuspidal edge on a focal surface using singularities of parallel surface of…

In this paper, we will interest in finding the number of zeros of the quadratic forms over finite fields. We will apply the tool for finding the number of rational points of supersingular curves in [6]. We will give some more tools for…

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…

In this paper we explore Kruyswijk's method and show how to obtain congruences for cubic partition. That apart we also examine inequalities for a(n) and provide upper bound for it in the fashion of the classic partition function p(n).

In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which…

This paper is concerned with the problem of finding $n$ distinct squares such that, on excluding any one of them, the sum of the remaining $n-1$ squares is a square. While parametric solutions are known when $n=3$ and $n=4$, when $n > 4$,…

In this article we establish some properties regarding the solutions of a linear congruence, bases of solutions of a linear congruence, and the finding of other solutions starting from these bases.

We investigate the distribution of rational points on singular cubic surfaces, whose coordinates have few prime factors. The key tools used are universal torsors, the circle method and results on linear equations in primes.

In the present note we study combinatorial and algebraic properties of cubic-line arrangements in the complex projective plane admitting nodes, ordinary triple and $A_{5}$ singular points. We deliver a Hirzebruch-type inequality for such…

We study surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and…

We introduce Kummer surfaces X=Km(CxC) with the group scheme G=mu_2 acting on the self-product of the rational cuspidal curve in characteristic two. The resulting quotients are normal surfaces having a configuration of sixteen rational…

We develop explicit techniques to investigate algebraic quasi-hyperbolicity of singular surfaces through the constraints imposed by symmetric differentials. We apply these methods to prove that rational curves on Barth's sextic surface,…

It is shown that the cubic nonconventional ergodic averages of any order with a bounded aperiodic multiplicative function or von Mangoldt weights converge almost surely.

In this paper, some particular rational maps P_n ---> P_n+1, called quadratic congruences, are studied. They appear in the theory of exceptional vector bundles on projective spaces.

Segre proved that a smooth cubic surface over Q is unirational iff it has a rational point. We prove that the result also holds for cubic hypersurfaces over any field, including finite fields.

Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with…

For each integer $k \in [0,9]$, we count the number of plane cubic curves defined over a finite field $\mathbb{F}_q$ that do not share a common component and intersect in exactly $k\ \mathbb{F}_q$-rational points. We set this up as a…

A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds or a wide class of surfaces over number fields for which…

In this paper, we continue the study of the relation between rational points of rational elliptic surfaces and plane curves. As an application, we give first examples of Zariski pairs of cubic-line arrangements that do not involve…