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A constructive version of Newton-Puiseux theorem for computing the Puiseux expansions of algebraic curves is presented. The proof is based on a classical proof by Abhyankar. Algebraic numbers are evaluated dynamically; hence the base field…
We determine all triples $(e,d,n)$ for which a general degree $d$ hypersurface $X\subset \mathbb{P}^n$ contains a degree $e$ rational curve $C$ with balanced restricted tangent bundle $T_X|_C$. In addition, we show how to compute explicit…
Ruled surfaces play an important role in various types of design, architecture, manufacturing, art, and sculpture. They can be created in a variety of ways, which is a topic that has been the subject of much discussion in mathematics and…
We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves…
We discuss a general growth curve including several parameters, whose choice leads to a variety of models including the classical cases of Malthusian, Richards, Gompertz, Logistic and some their generalizations. The advantage is to obtain a…
Consider the scheme parametrizing non-constant morphisms from a fixed projective curve to a projective surface. There is a rational map between this scheme and the Chow variety of $1$-cycles on the surface. We prove that, if the curve is…
We study projective curves and hypersurfaces defined over a finite field that are tangent to every member of a class of low-degree varieties. Extending 2-dimensional work of Asgarli, we first explore the lowest degrees attainable by smooth…
We generalize results by Wakabayashi and Orevkov about rational cuspidal curves on the projective plane to that on $\mathbb{Q}$-homology projective planes. It turns out that the result is exactly the same as the projective plane case under…
We study negative curves on surfaces obtained by blowing up special configurations of points in the complex projective palne. Our main results concern the following configurations: very general points on a cubic, 3-torsion points on an…
We classify linearly normal surfaces $S \subset \mathbf{P}^{r+1}$ of degree $d$ such that $4g-4 \leq d \leq 4g+4$, where $g>1$ is the sectional genus (it is a classical result that for larger $d$ there are only cones). We apply this to the…
We give a novel and effective criterion for algebraicity of rational normal analytic surfaces constructed from resolving the singularity of an irreducible curve-germ on $CP^2$ and contracting the strict transform of a given line and all but…
We generalize the work of Dem'janenko and Silverman for the Fermat quartics, effectively determining the rational points on the curves $x^{2m}+ax^m+ay^m+y^{2m}=b$ whenever the ranks of some companion hyperelliptic Jacobians are at most one.…
Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two…
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…
A telegraphic survey of some of the standard results and conjectures about the set $C({\bf Q})$ of rational points on a smooth projective absolutely connected curve $C$ over ${\bf Q}$.
This article aims to extend classical homological results about the rational normal curves to analogues in weighted projective spaces. Results include determinantality and nonstandard versions of quadratic generation and the Koszul…
We generalize a previous result by Fabricius-Bjerre from curves in $\mathbb R^2$ to curves in $\mathbb R P^2$. Applied to the case of real algebraic curves, this recovers the signed count of bitangents of quartics introduced by Larson-Vogt…
This work consists of two parts. In the first part we develop new techniques to compute Koszul cohomology groups for several classes of varieties. As applications we prove results on projective normality and syzygies for algebraic surfaces.…
We prove new theorems which are higher-dimensional generalizations of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from $\mathbb{C}$ to affine curves. These include results on…
This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this…