Irreducibility of equisingular families of curves

In 1985 Joe Harris proved the long-standing claim of Severi that equisingular families of plane nodal curves are irreducible whenever they are nonempty. For families with more complicated singularities this is no longer true. Given a divisor on a smooth projective surface it thus makes sense to look for conditions which ensure that the family of irreducible curves in the linear system with precisely singular points of types is irreducible. Considering different surfaces, including general surfaces in and products of curves, we produce a sufficient condition of the type

where is some constant and some zero-dimensional scheme associated to the singularity type. Our results carry the same asymptotics as the best known results in this direction in the plane case, even though the coefficient is worse. For most of the surfaces considered these are the only known results in that direction.

     

Smoothness of equisingular families of curves

Francesco Severi (1921) showed that equisingular families of plane nodal curves are T-smooth, i.e. smooth of the expected dimension, whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a divisor on a smooth projective surface it thus makes sense to look for conditions which ensure that the family of irreducible curves in the linear system with precisely singular points of types is T-smooth. Considering different surfaces including the projective plane, general surfaces in , products of curves and geometrically ruled surfaces, we produce a sufficient condition of the type

where is some invariant of the singularity type and is some constant. This generalises the results of Greuel, Lossen, and Shustin (2001) for the plane case, combining their methods and the method of Bogomolov instability. For many singularity types the -invariant leads to essentially better conditions than the invariants used by Greuel, Lossen, and Shustin (1997), and for most classes of geometrically ruled surfaces our results are the first known for T-smoothness at all.

     

Lattice polygons and families of curves on rational surfaces

First we solve the problem of finding minimal degree families on toric surfaces by reducing it to lattice geometry. Then we describe how to find minimal degree families on, more generally, rational complex projective surfaces.     

Focal loci of families and the genus of curves on surfaces

In this article we apply the classical method of focal loci of families to give a lower bound for the genus of curves lying on general surfaces. First we translate and reprove Xu's result that any curve on a general surface in of degree has geometric genus . Then we prove a similar lower bound for the curves lying on a general surface in a given component of the Noether-Lefschetz locus in and on a general projectively Cohen-Macaulay surface in .

     

A degree bound for families of rational curves on surfaces

We give an upper bound for the degree of rational curves in a family that covers a given birationally ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We introduce an algorithm for constructing examples where the upper bound is tight. As an application of our methods we improve an inequality on lattice polygons.     

Self-intersections of curves on surfaces

Let M be a compact orientable surface with nonempty boundary (x(M)<0) and fundamental group . Let be a geodesic on M (with a fixed hyperbolic structure), and let W be a (cyclically reduced) word in a fixed set of generators of which represents . In this paper, we give an algorithm to count the number of self-intersections of in terms of W, generalizing a result of Birman and Series, where an algorithm was given to decide if was simple. Some applications of the algorithm to surfaces with one boundary and the Markoff spectrum are also given.     

Intersections of curves on surfaces

The authors consider curves on surfaces which have more intersections than the least possible in their homotopy class. Theorem 1.Let f be a general position arc or loop on an orientable surface F which is homotopic to an embedding but not embedded. Then there is an embedded 1-gon or 2-gon on F bounded by part of the image of f. Theorem 2.Let f be a general position arc or loop on an orientable surface F which has excess self-intersection. Then there is a singular 1-gon or 2-gon on F bounded by part of the image of f. Examples are given showing that analogous results for the case of two curves on a surface do not hold except in the well-known special case when each curve is simple.     

Finite sets on curves and surfaces

A complete proof is given for Schnirelmann’s theorem on the existence of a square inC ² Jordan curves. The following theorems are then proved, using the same method: 1. On every hypersurface inR ⁿ,C ³-diffeomorphic toS ⁿ⁻¹, there exist 2n points which are the vertices of a regular 2 ⁿ -cellC _n. 2. Every planeC′ Jordan curve can beC′ approximated by a curve on which there are 2N distinct points which are the vertices of a centrally symmetric 2N-gon (anglesπ not excluded). 3. On every planeC ² curve there exist 5 distinct points which are the vertices of an axially symmetric pentagon with given base anglesa, π/2≦a<π. (The angle at the vertex on the axis of symmetry might beπ). Research supported by Grant AF-AFOSR-664-64, Air Force Office of Scientific Research.     

Integer points on curves and surfaces

Various upper bounds are given for the number of integer points on plane curves, on surfaces and hypersurfaces. We begin with a certain class of convex curves, we treat rather general surfaces in ³ which include algebraic surfaces with the exception of cylinders, and we go on to hypersurfaces in ⁿ with nonvanishing Gaussian curvature.Written with partial supports from NSF grant No. MCS-8211461.     

Counting curves on rational surfaces

In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface. Received: 30 June 1999 / Revised version: 1 January 2000     

Isoperimetric curves on hyperbolic surfaces

Least-perimeter enclosures of prescribed area on hyperbolic surfaces are characterized.

     

Parameter spaces for curves on surfaces and enumeration of rational curves

Let S be a smooth, minimal rational surface. The geometry of the Severi variety parametrising irreducible, rational curves in a given linear system on S is studied. The results obtained are applied to enumerative geometry, in combination with ideas from Quantum Cohomology. Formulas enumerating rational curves are found, some of which generalised Kontsevich's formula for plane curves.     

Remarks on infinitesimally desarguesian families of curves

Infinitesimally Desarguesian two-parameter families of curves in the plane which are in a sense close to the family of straight lines are discussed. Their properties, examples, and multidimensional generalizations are considered.     

Strebel differentials on families of hyperelliptic curves

Our paper is devoted to Strebel pairs on families of hyperelliptic curves. We provide a complete proof of the fact that the constructed differentials are Strebel and point out the connections between the introduced constructions and classical objects of complex algebraic geometry. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 6, pp. 121–130, 2007.     

A note on families of hyperelliptic curves

We give a stack-theoretic proof for some results on families of hyperelliptic curves. Received: 5 February 2008