The gonality sequence of a curve with an involution

If a curve X of genus g is a double covering of a curve C of genus h such that g ≥ 6h?3 ≥ 9, there is an explicit relation between the gonality sequences of X and C. In particular, it shows that X violates the slope inequalities if and only if C does. This provides new examples of curves X violating the slope inequalities.     

On the gonality sequence of smooth curves

Let C be a smooth curve of genus g. For each positive integer r the r-gonality d _r (C) of C is the minimal integer t such that there is \({L\in {\rm Pic}^t(C)}\) with h ⁰(C, L) = r + 1. Here we use nodal plane curves to construct several smooth curves C with d ₂(C)/2 < d ₃(C)/3, i.e., for which a slope inequality fails.     

The separating gonality of a separating real curve

Let X be a smooth real curve of genus g such that the real locus has s connected components. We say X is separating if the complement of the real locus is disconnected. In case there exists a morphism f from X to ${\mathbb{P}^1}$ such that the inverse image of the real locus of ${\mathbb{P}^1}$ is equal to the real locus of X then X is separating and such morphism is called separating. The separating gonality of a separating real curve X is the minimal degree of a separating morphism from X to ${\mathbb{P}^1}$ . It is proved by Gabard that this separating gonality is between s and (g + s + 1)/2. In this paper we prove that all values between s and (g + s + 1)/2 do occur.     

Even points on an algebraic curve

We show that the points of a global function field, whose classes are 2-divisible in the Picard group, form a connected regular infinite graph, with the incidence relation generalizing the well known quadratic reciprocity law. We prove that for every global function field the diameter of this graph is precisely 2. In addition we develop an analog of global square theorem that concerns points 2-divisible in the Picard group.     

On the gonality of graph curves

A genus g graph curve in the sense of Bayer and Eisenbud is a genus g stable curve (hence nodal) with 2g − 2 irreducible components, each of then smooth and rational and intersecting exactly 3 other components. Here, we study the existence of spanned or very ample non-special line bundles on X whose restriction to each irreducible component of X has degree 1.     

Homological equivalence relations on an algebraic curve

We study the collection of homological equivalence relations on a fixed curve. We construct a moduli space for pairs consisting of a curve of genus g and a homological equivalence relation of degree n on the curve, and a classifying set for homological equivalence relations of degree n on a fixed curve, modulo automorphisms of the curve. We identify a special type of homological equivalence relations, and we characterize the special homological equivalence relations in terms of the existence of elliptic curves in the Jacobian of the curve.     

On the gonality of nodal curves

Here we prove that for every n33 and every t(n ²+3n)/6, the normalization Y of a general plane curve C of degree n and with t nodes has no g _b ¹ with b<n–2 and only g _n–2 ¹ and g _n–1 ¹ induced by a pencil of lines through a point of C. Recently, Coppens and Kato proved stronger results.     

On the gonality of stable curves

In this paper we use admissible covers to investigate the gonality of a stable curve C over $\mathbb{C}$. If C is irreducible, we compare its gonality to that of its normalization. If C is reducible, we compare its gonality to that of its irreducible components. In both cases we obtain lower and upper bounds. Furthermore, we show that four admissible covers constructed give rise to generically injective maps between Hurwitz schemes. We show that the closures of the images of three of these maps are components of the boundary of the target Hurwitz schemes, and the closure of the image of the remaining map is a component of a certain codimension‐1 subscheme of the boundary of the target Hurwitz scheme.     

On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

Let M(n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M(n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms ${M or_P (\mathbb{G}, M(n, \xi))}$ and the moduli space of stable bundles over ${X \times \mathbb{G}}$ , where ${\mathbb{G}}$ is the Grassmannian ${\mathbb{G}(n - r, \mathbb{C}^n)}$ . Moreover, we give sufficient conditions for ${M or_{2ns}(\mathbb{P}^1, M(n, \xi))}$ to be non-empty, when s ≥ 1.     

Estimation of a sum along an algebraic curve

Let be an algebraic curve determined over a finite field k = [q]; e,x are subsidiary additive and multiplicative characters of the field k;, are functions in determined over k and satisfying some natural conditions. If P passes through the points of curve , rational over k, then where constant C depends only on the powers of ,,.Translated from Matematicheskie Zametki, Vol. 5, No. 3, pp. 373–380, March, 1969.