On the distribution of Weierstrass points on singular curves

Weierstrass points are defined for invertible sheaves on integral, projective Gorenstein curves. An example is given of a rational nodal curveX and an invertible sheaf ℒ of positive degree onX such that the set of all higher order Weierstrass points of ℒ is not dense inX.     

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.

     

The Weierstrass semigroup of a pair of GaloisWeierstrass points with prime degree on a curve

We describe the Weierstrass semigroup of a Galois Weierstrass point with prime degree and the Weierstrass semigroup of a pair of Galois Weierstrass points with prime degree, where a Galois Weierstrass point with degree n means a total ramification point of a cyclic covering of the projective line of degree n.*Supported by Korea Research Foundation Grant (KRF-2003-041-C20010).**Partially supported by Grant-in-Aid for Scientific Research (15540051), JSPS.     

On the index of the Weierstrass semigroup of a pair of points on a curve

We obtain the exact formulas for the cardinality of the complement of the Weierstrass semigroup of a pair (p, q) of points on a curveC. Using these formulas we obtain lower bounds and upper bounds on the cardinalities of such sets. Moreover, considering examples, we show that these bounds are sharp.Partially supported by Korea Science and Engineering Foundation and by Global Analysis Research Center.     

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.     

On the monodromy of weierstrass points

Summary In this paper we consider the family of curves of genus g=2m with a g ₃ ¹ lying on a particular rational normal scroll S in P^g– 1(C). We define a covering of this family representing the Weierstrass points and we study the monodromy. Applying the techniques of [3] we prove that if g=4 the monodromy is the full symmetric group and for general g=2m it is transitive. We show also that the generic curve of the family has only normal Weierstrass points generalizing a classical result. We work always over the complex numbers.Partially supported by: Ministero della Pubblica Istruzione - Italia; Consiglio Nazionale delle Ricerche — Italia.     

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 certain curves of genus three in characteristic two

We study curves of genus 3 over algebraically closed fields of characteristic 2 with the canonical theta characteristic totally supported in one point. We compute the moduli dimension of such curves and focus on some of them which have two Weierstrass points with Weierstrass directions towards the support of the theta characteristic. We answer questions related to order sequence and Weierstrass weight of Weierstrass points and the existence of other Weierstrass points with similar properties. – Dedicated to the treasured memory of our coauthor, Paulo Henrique Viana     

Ray class fields generated by torsion points of certain elliptic curves

We first normalize the derivative Weierstrass ???-function appearing in the Weierstrass equations which give rise to analytic parametrizations of elliptic curves, by the Dedekind ??-function. And, by making use of this normalization of ???, we associate a certain elliptic curve to a given imaginary quadratic field K and then generate an infinite family of ray class fields over K by adjoining to K torsion points of such an elliptic curve. We further construct some ray class invariants of imaginary quadratic fields by utilizing singular values of the normalization of ???, as the y-coordinate in the Weierstrass equation of this elliptic curve, which would be a partial result towards the Lang?CSchertz conjecture of constructing ray class fields over K by means of the Siegel?CRamachandra invariant.     

Local invariants attached to Weierstrass points

Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation d of the hyperelliptic discriminant of X/S, and the valuation δ of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable the invariants d and δ are known to satisfy certain inequalities. We prove an exact formula relating d and δ with intersection theoretic data determined by the distribution of Weierstrass points over the special fiber, in the semistable case. We also prove an exact formula for the stable Faltings height of an arbitrary curve over a number field, involving local contributions associated to its Weierstrass points.     

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.     

Families of special Weierstrass points

The purpose of this Note is to show that loci of (special) Weierstrass points on the fibers of a family $\pi :\mathfrak{X}\to S$ of smooth curves of genus $g?2$ can be studied by simply pulling back the Schubert calculus naturally living on a suitable Grassmann bundle over $\mathfrak{X}$. Using such an idea we prove new results regarding the decomposition in $A_1(\mathfrak{X})$ of the class of the locus of Weierstrass points having weight at least 3 as the sum of classes of Weierstrass points having “bounded from below” gaps sequences. To cite this article: L. Gatto, P. Salehyan, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

The group of Weierstrass points of a plane quartic with at least eight hyperflexes

The group generated by the Weierstrass points of a smooth curve in its Jacobian is an intrinsic invariant of the curve. We determine this group for all smooth quartics with eight hyperflexes or more. Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed number of hyperflexes in the moduli space of curves of genus 3.

     

The monodromy of Weierstrass points

Summary We show that the monodromy of the family of curves (Riemann surfaces) acts as the full symmetric group on the Weierstrass points of a general curve. The proof uses a degeneration to certain reducible curves, and the theory of limit series developed in our (1986, 1987a, b). Some of the monodromy is actually constructed by fixing a (reducible) curve and varying its canonical series.Both authors are grateful to the National Science Foundation for partial support during the preparation of this work     

Intersection Divisors of a Canonically Embedded Curve with its Osculating Spaces

We present a new result on the geometry of nonhyperelliptic curves; namely, the intersection divisors of a canonically embedded curve C with its osculating spaces at a point P, not considering the intersection at P, can only vary in dimensions given by the Weierstrass semigroup of the curve C at P. We obtain, under a reasonable geometrical hypothesis, monomial bases for the spaces of higher-order regular differentials. We also give a sufficient condition on the Weierstrass semigroup of C at P in order for this geometrical hypothesis to be true. Finally, we give examples of Weierstrass semigroups satisfying this condition.     

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.     

On the constellations of Weierstrass points

We prove that the constellation of Weierstrass points characterizes the isomorphism-class of double coverings of curves of genus large enough. The author was supported by a grant from the International Atomic Energy Agency and UNESCO.     

Weierstrass points and their impact in the study of algebraic curves: a historical account from the “Lückensatz” to the 1970s

In this note we give a historical account of the origin and the development of the concept of Weierstrass point. We also explain how Weierstrass points have contributed to the study of compact Riemann surfaces and algebraic curves in the century from Weierstrass’ statement of the gap theorem to the 1970s. In particular, we focus on the seminal work of Hürwitz that raised questions which are at the center of contemporary research on this topic.      

On curves with all Weierstrass points of maximal weight

In this note we show that the few known examples of non-hyperelliptic complex algebraic curves all of whose Weierstrass points have maximal weight are the only ones.