Related papers: The end of the curve complex
Working over imperfect fields, we give a comprehensive classification of genus-one curves that are regular but not geometrically regular, extending the known case of geometrically reduced curves. The description is given intrinsically, in…
A graph $G$ embedded in a surface $S$ is called an $S$-grid when every facial boundary walk has length four, that is, the topological dual graph of $G$ in $S$ is 4-regular. Aside from the case where $S$ is the torus or Klein bottle, an…
We exhibit planar, rational curves of large degree over ${\mathbb F}_2$ that have a unique singular point, which has multiplicity 2. In characteristic 0 such curves exist only for degrees up to $6$. v.2: references updated and examples of…
It is well known that there exist knots with Seifert surfaces of arbitrarily high genus. In this paper, we show the existence of infinitely many knot exteriors where each of which has longitudinal essential surfaces of any positive genus…
We study simple Osserman limit linear series (that is, Osserman limit linear series having a simple basis) on curves of compact type with three irreducible components. For compact type curves with two components, every exact limit linear…
A smooth real curve is called separating in case the complement of the real locus inside the complex locus is disconnected. This is the case if there exists a morphism to the projective line whose inverse image of the real locus of the…
It is shown that various questions about the existence of simple closed curves in normal subgroups of surface groups are undecidable.
Suppose C is a singular curve in CP^2 and it is topologically an embedded surface of genus g; such curves are called cuspidal. The singularities of C are cones on knots K_i. We apply Heegaard Floer theory to find new constraints on the sets…
The complex of domains $D(S)$ is a geometric tool with a very rich simplicial structure, it contains the curve complex $C(S)$ as a simplicial subcomplex. In this paper we shall regard it as a metric space, endowed with the metric which…
The matching complex of a graph is the simplicial complex whose vertex set is the set of edges of the graph with a face for each independent set of edges. In this paper we completely characterize the pairs (graph, matching complex) for…
Let C_g be a general curve of genus g>3. Guralnick and others proved that the monodromy group of a cover C_g-> P^1 of degree n is either S_n or A_n. We show that A_n occurs for n>2g. The corresponding result for S_n is classical.
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…
In this paper a new intrinsic geometric characterization of the symmetric square of a curve and of the ordinary product of two curves is given. More precisely it is shown that the existence on a surface of general type S of irregularity q…
Consider the Jacobian of a genus two curve defined over a finite field and with complex multiplication. In this paper we show that if the l-Sylow subgroup of the Jacobian is not cyclic, then the embedding degree of the Jacobian with respect…
The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre upper bound for the number of points on the curve. We present algorithms for constructing curves of genus 5,…
We find a geometrical method of analysing the singularities of a plane nodal curve. The main results will be used in a forthcoming paper on geometric Plucker formulas for such curves. Plane nodal curves, that is plane curves having at most…
It is proved that the rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. As an illustration, we consider a family of elliptic curves with complex multiplication.
Let $C$ be a genus one nodal curve over a local artinian base and let $E$ be a proper subcurve of genus one. We define residues for curves over local artinian rings, then define generalized residues with values in line bundles over the…
A simple graph is called triangular if every edge of it belongs to a triangle. We conjecture that any graphical degree sequence all terms of which are greater than or equal to 4 has a triangular realisation, and establish this conjecture…
We propose a topological approach to the problem of determining a curve from its iterated integrals. In particular, we prove that a family of terms in the signature series of a two dimensional closed curve with finite p variation, 1\leq…