Related papers: On Enriques-Babbage Theorem for singular curves

Let $L$ be a line bundle on a K3 or Enriques surface. We give a vanishing theorem for $H^1(L)$ that, unlike most vanishing theorems, gives necessary and sufficient geometrical conditions for the vanishing. This result is essential in our…

In this article we produce Groebner bases for the defining ideal of a monomial curve that corresponds to an almost arithmetic sequence of positive integers, correcting previous work of Sengupta,(2003).

Let $C$ be an integral and projective curve; and let $C'$ be its canonical model. We study the relation between the gonality of $C$ and the dimension of a rational normal scroll $S$ where $C'$ can lie on. We are mainly interested in the…

Let $C$ be an irreducible projective plane curve in the complex projective space ${\mathbb{P}}^2$. The classification of such curves, up to the action of the automorphism group $PGL(3,{\mathbb{C}})$ on ${\mathbb{P}}^2$, is a very difficult…

We bound the maximal number N of singular points of a plane algebraic curve C that has precisely one place at infinity with one branch in terms of its first Betti number $b_1(C)$. Asymptotically we prove that $N<\sim{17/11}b_1(C)$ for large…

We use Morse theoretical arguments to study algebraic curves in C^2. We take an algebraic curve C in C^2 and intersect it with a family of spheres with fixed origin and varying radii. We explain in detail how does the resulting link change…

Motivated by the analogy between number fields and function fields, this paper extends the main result of \cite{janbazi2025unified} to the function field setting. Let $C$ be a smooth affine curve over a finite field, and let $\pi: S…

We give a method to construct explicitly a supersingular curve of given genus g in characteristic 2.

Let E be a plane rational curve defined over complex numbers which has only locally irreducible singularities. The Coolidge-Nagata conjecture states that E is rectifiable, i.e. it can be transformed into a line by a birational automorphism…

The automorphism group of a curve is studied from the viewpoint of the canonical embedding and Petri's theorem. A criterion for identifying the automorphism group as an algebraic subgroup the general linear group is given. Furthermore the…

In this paper a new classification of monomial curves in $\mathbb{A}^4(\mathbbmss{k})$ is given. Our classification relies on the detection of those binomials and monomials that have to appear in every system of binomial generators of the…

We classify singular Enriques surfaces in characteristic two supporting a rank nine configuration of smooth rational curves. They come in one-dimensional families defined over the prime field, paralleling the situation in other…

In this paper, we study the deformation theory of degenerate algebraic curves on singular varieties which appear as the degenerate limit of families of varieties. For this purpose, we systematically develop a new method to calculate the…

We give a new proof of the Alexander-Wermer Theorem that characterizes the oriented curves in C^n which bound positive holomorphic chains, in terms of the linking numbers of the curve with algebraic cycles in the complement. In fact, we…

It is a very old and interesting open problem to characterize those collections of embedded topological types of local plane curve singularities which may appear as singularities of a projective plane curve C of degree d. The goal of the…

We prove an extension of the Babbage-Enriques-Petri theorem for semi-canonical curves. We apply this to show that the Prym variety of a generic element of a codimension $k$ subvariety of $\mathcal{R}_{g}$ is not isogenous to another…

A new generalization of the classical separate algebraicity theorem is suggested and proved.

A curve C in the projective plane is called non-negative if the self-intersection number of C after the minimal resolution of singularities of C is non-negative. Given a unicuspidal rational plane curve C with singular point P, we study the…

Green's Conjecture states the following : syzygies of the canonical model of a curve are simple up to the p^th stage if and only if the Clifford index of C is greater than p. We prove that the generic curve of genus g satisfies Green's…

In [3], it is proved that the quotient of an abelian variety $A$ by a finite order automorphism $g$ is uniruled if and only if some power of $g$ satisfies a numerical condition $0<\age(g^k)<1$. In this paper, we show that $\age(g^k)=1$ is…