Related papers: On Enriques-Babbage Theorem for singular curves

It is proved that special cubic B\'ezier curves, generated from quadratic curves by the use of a scalar parameter, have at most one local curvature extremum in the $(0,1)$ parameter interval.

Let $C$ be a compact complex curve included in a non-singular complex surface such that the normal bundle is topologically trivial. Ueda studied complex analytic properties of a neighborhood of $C$ when $C$ is non-singular or is a rational…

In this paper we introduce the notion of the singular evolutoid set which is the set of all singular points of all evolutoids of a fixed smooth planar curve with at most cusp singularities. By the Gauss-Bonnet Theorem for Coherent Tangent…

Baker's method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we give a generalisation of…

We give conceptual proofs of some results on the automorphism group of an Enriques surface X, for which only computational proofs have been available. Namely, there is an obvious upper bound on the image of Aut(X) in the isometry group of…

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…

We present a uniqueness theorem for k-graph C*-algebras that requires neither an aperiodicity nor a gauge invariance assumption. Specifically, we prove that for the injectivity of a representation of a k-graph C*-algebra, it is sufficient…

Let $E$ be an elliptic curve over a number field $K$ with at least one real embedding and $L$ be a finite extension of $K$. We generalize a result of Habegger to show that $L(E_{\text{tor}})$, the field generated by the torsion points of…

We describe a model-theoretic setting for the study of Shimura varieties, and study the interaction between model theory and arithmetic geometry in this setting. In particular, we show that the model-theoretic statement of a certain…

We discuss $p$-adic unipotent Albanese maps for curves of positive genus, extending the theory of $p$-adic multiple polylogarithms. This construction is then used to relate linear Diophantine conjectures of `Birch and Swinnerton-Dyer type'…

In this paper we provide some factorization theorems of the Poincar\'e series $P_C$ of a plane curve singularity $C$ depending on some key values of the semigroup of values of \(C\). These results yield an iterative computation of $P_C$ in…

In this paper, Legendre curves on unit tangent bundle are given using rotation minimizing (RM) vector fields. Ruled surfaces corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and…

We show some of the conjectures of Pappas and Rapoport concerning the moduli stack of $\mathcal{G}$-torsors on a curve C, where $\mathcal{G}$ is a semisimple Bruhat-Tits group scheme on C. In particular we prove the analog of the…

In this paper, we are interested in the generic initial ideals of \textit{singular} projective curves with respect to the graded lexicographic order. Let $C$ be a \textit{singular} irreducible projective curve of degree $d\geq 5$ with the…

Let $\mathcal C$ be a real plane algebraic curve defined by the resultant of two polynomials (resp. by the discriminant of a polynomial). Geometrically such a curve is the projection of the intersection of the surfaces $P(x,y,z)=Q(x,y,z)=0$…

This is a brief introduction to the theory of Enriques surfaces over arbitrary algebraically closed fields. Some new results about automorphism groups of Enriques surfaces are also included.

Let A be an abelian variety defined over a number field F. For a prime number $\ell$, we consider the field extension of F generated by the $\ell$-powered torsion points of A. According to a conjecture made by Rasmussen and Tamagawa, if we…

Let C be an algebraic curve of genus g. Consider extensions E of a vector bundle F'' of rank n'' by a vector bundle F' of rank n'. The following statement was conjectured by Lange: If 0<n'deg F''-n''degF'\le n'n''(g-1), then there exist…

We prove results generalizing the classical Riemann Singularity Theorem to the case of integral, singular curves. The main result is a computation of the multiplicity of the theta divisor of an integral, nodal curve at an arbitrary point.…

Let $C$ be a nodal curve, and let $E$ be a union of semistable subcurves of $C$. We consider the problem of contracting the connected components of $E$ to singularities in a way that preserves the genus of $C$ and makes sense in families,…