Weierstrass multiple points on algebraic curves and ramified coverings

Here we prove the following result on Weierstrass multiple points. Theorem:Fix integers k, g with k≥5 and g>4k. Then there exist a genus g, Riemann surface X and k points P ₁, …,P _k of X such that for all integers b ₁≥…≥b _k ≥0we have:

On Weierstrass semigroups of double coverings of genus three curves

Let C be a complete non-singular curve of genus 3 over an algebraically closed field of characteristic 0. We determine all possible Wierstrass semigroups of ramification points on double coverings of C whose covering curves have genus greater than 8. Moreover, we construct double coverings with ramification points whose Weierstrass semigroups are the possible ones.     

Weierstrass points on certain non-classical curves

Archiv der Mathematik -     

Weierstrass multiple points and ramification points of smooth projective curves

Summary In this paper we study finite sets of smooth algebraic curves which are the support of special divisors («Weierstrass sets»). We prove several existence results of Weierstrass sets with low weight on suitable curves (e.g. general k-gonal curves).     

On double coverings of hyperelliptic curves

We prove various properties of varieties of special linear systems on double coverings of hyperelliptic curves. We show and determine the irreducibility, generically reducedness and singular loci of the variety for bi-elliptic curves and double coverings of genus two curves. Similar results for double coverings of hyperelliptic curves of genus h≥3 are also presented.     

Pencils on double coverings of curves

Let X be a smooth curve of genus g. When and d ≥ π−2g+1 we show the existence of a double covering where C a smooth curve of genus π with a base-point-free pencil of degree d which is not the pull-back of a pencil on X. Received: 7 February 2007; Revised: 1 July 2008     

Limit Weierstrass points on nodal reducible curves

In the 1980s D. Eisenbud and J. Harris posed the following question: ``What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type?' In the present article, we give a partial answer to this question. We consider the case where the limit curve has components intersecting at points in general position and where the degeneration occurs along a general direction. For this case we compute the limits of Weierstrass points of any order. However, for the usual Weierstrass points, of order one, we need to suppose that all of the components of the limit curve intersect each other.

     

A characterization of double coverings of curves

The gonality sequence \({(d_{r})_{r}}\) of a curve X of genus g which doubly covers a curve of genus h satisfies \({d_{r} = 2(r + h)}\) for all \({r = h, h + 1, \ldots, g - 3h}\) provided that \({g \gg h}\). In this paper we explore if this striking feature of \({(d_{r})_{r}}\) actually characterizes such a covering.     

Weierstrass points on trigonal curves I: The ramification points

LetC be a smooth curve of genusg≥5. Assume thatP is a Weierstrass point onC which first non-gap is equal to 3. The gap sequence atP is completely determinated by numbersn and ε satisfying (g−1)/3≤n≤g/2 and ε is 1 or 2 as follows. Given suchn and ε, the corresponding gap sequence is (1, 2, 4, 5,…, 3n−2, 3n−1, 3n+ε, 3n+3+ε, …, 3(g−n−1)+ε). We say thatP is of then-th kind andP is of type I (resp. II) if ε=1 (resp. 2). Because a curve of genusg≥5 has at most one linear systemg1/3, it follows that the Weierstrass points onC with first non-gap equal to 3 are of the same kind.     

Weierstrass points and gap sequences for families of curves

The theory of Weierstrass points and gap sequences for linear series on smooth curves is generalized to smooth families of curves with geometrically irreducible fibers, and over an arbitrary base scheme.     

Linear systems on curves with no Weierstrass points

We study order-sequences of linear systems on smooth curves and establish the formula:b _j +b _N−j ≤b _N for allj, where {b ₀<b ₁<...<b _N } is the order-sequence of a linear system on a curve. As an application of the formula, we describe all linear systems on curves which have no Weierstrass points.     

Rational nodal curves with no smooth Weierstrass points

Let denote the rational curve with nodes obtained from the Riemann sphere by identifying 0 with and with for , where is a primitive th root of unity. We show that if is even, then has no smooth Weierstrass points, while if is odd, then has smooth Weierstrass points.