Related papers: Totally real divisors on curves

As proved recently in [PT], for varieties $X^{r+1}\subset \mathbb P^N$ such that through $n\geq 2$ general points there passes an irreducible curve $C$ of degree $\delta\geq n-1$ we have $N\leq \pi(r,n,\delta+r(n-1)+2)$, where $\pi(r,n,d)$…

We produce an integral model for the modular curve $X(Np^m)$ over the ring of integers of a sufficiently ramified extension of $\mathbf{Z}_p$ whose special fiber is a {\em semistable curve} in the sense that its only singularities are…

We study the finiteness of low degree points on certain modular curves and their Atkin--Lehner quotients, and, as an application, prove the modularity of elliptic curves over all but finitely many totally real fields of degree $5$. On the…

Let $k$ be a perfect field of characteristic $p>0$, and let $d$ be a positive integer not divisible by $p$. We define a non-empty Zariski open subset $U$ of the space of polynomials of degree $d$, and for $f(x)\in U(k)$, we compute the…

Classification of curves in a projective space occupies minds of many mathematicians. First step in doing so is classification of curves on a given surface. This brings us to consideration of the nonsingular Del Pezzo Surface in $P^4_k.$ We…

This text is a presentation of a set of formulae, first found by Vainsencher (for $\delta \leq 6$) and shortly after improved by Kleiman and Piene, counting $\delta$-nodal curves in a complete linear system on a smooth surface, if $\delta…

We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…

It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and…

Let $(G, \omega)$ be a hyperelliptic vertex-weighted graph of genus $g \geq 2$. We give a characterization of $(G, \omega)$ for which there exists a smooth projective curve $X$ of genus $g$ over a complete discrete valuation field with…

We study special linear systems called "very special" whose dimension does not satisfy a Clifford type inequality given by Huisman. We classify all these very special linear systems when they are compounded of an involution. Examples of…

Let K be the function field of a connected regular scheme S of dimension 1, and let f : X -> Y be a finite cover of projective smooth and geometrically connected curves over K with g(X) greater or equal to 2. Suppose that f can be extended…

In this paper we investigate complex uniruled varieties $X$ whose rational curves of minimal degree satisfy a special property. Namely, we assume that the tangent directions to such curves at a general point $x\in X$ form a linear subspace…

Following the work of Totaro and Pereira, we study sufficient conditions under which collections of pairwise-disjoint divisors on a variety over an algebraically closed field are contained in the fibers of a morphism to a curve. We prove…

We study equisingularity of families of reduced curves over smooth parameter spaces of arbitrary positive dimension, using the difference between two analytic invariants of a curve singularity: the multiplicity of its Jacobian ideal and its…

In this paper we study the embedded topology of reducible plane curves having a smooth irreducible component. In previous studies, the relation between the topology and certain torsion classes in the Picard group of degree zero of the…

Typical existence result on Ricci-flat metrics is in manifolds of finite geometry, that is, on $F=\bar F-D$ where $\bar F$ is a compact K\"ahler manifold and $D$ is a smooth divisor. We view this existence problem from a different…

The $X$-rank of a point $p$ in projective space is the minimal number of points of an algebraic variety $X$ whose linear span contains $p$. This notion is naturally submultiplicative under tensor product. We study geometric conditions that…

Let n > 2 and let d < (n+1)/2. We prove that for a general hypersurface X of degree d in P^n, all the genus 0 Kontsevich moduli spaces M_{0,n}(X,e) are irreducible, reduced, local complete intersection stacks of the expected dimension.

Let $f \colon X \to B$ be a complex elliptic surface and let $\DD \subset X$ be an integral divisor dominating $B$. It is well-known that the Parshin-Arakelov theorem implies the Mordell conjecture over complex function fields by a…

We present a new, far simpler family of counter-examples to Kushnirenko's Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the…