Deformations of exceptional Weierstrass points (original) (raw)
Related papers
2010
Using first order deformation theory of pointed curves we show that the semigroup of a generic Weierstrass point whose semigroup has first nonzero element k consists only of multiples of k until after its greatest gap value, and that on the moduli space of curves two components of the divisor of points corresponding to curves possessing exceptional Weierstrass points intersect nontransversely. As usual let J( be the moduli space of curves of genus g. Many subloci of Jtg defined in terms of Weierstrass points have received much study. (1) Definition. For an integer k with 2 < k < g, Dk k = {[C] e Jfg. C possesses a Weierstrass point p with h°(C, kp) > 2). It is known that Dk k is irreducible of dimension 2g 3 + k and a general point in Dk k corresponds to a curve C with a Weierstrass point p with h°(C,kp) = 2. See Rauch [1], Lax [1, 2], Arbarello [1, 2]. Here we will prove: (2) Theorem. A generic point in Dk k corresponds to a curve with a Weierstrass point whose semigroup c...
On the distribution of Weierstrass points on irreducible rational nodal curves
Pacific Journal of Mathematics, 1990
Let X be an irreducible rational nodal curve of arithmetic genus g > 2, and let S* be a non-special, effective invertible sheaf on X. Let W(= S ?) denote the set of smooth Weierstrass points of 3? and all its positive tensor powers on I. In this paper, we study the distribution of W{3?) on X. In particular, we will show that is not dense on X, generalizing an example of R. F. Lax.
Weierstrass points and moduli of curves
Compositio Mathematica, 1974
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Torsion points of small order on hyperelliptic curves
European Journal of Mathematics, 2022
Let C be a hyperelliptic curve of genus g > 1 over an algebraically closed field K of characteristic zero and O one of the (2g + 2) Weierstrass points in C(K). Let J be the jacobian of C, which is a g-dimensional abelian variety over K. Let us consider the canonical embedding of C into J that sends O to the zero of the group law on J. This embedding allows us to identify C(K) with a certain subset of the commutative group J(K). A special case of the famous theorem of Raynaud (Manin-Mumford conjecture) asserts that the set of torsion points in C(K) is finite. It is well known that the points of order 2 in C(K) are exactly the "remaining" (2g + 1) Weierstrass points. One of the authors [12] proved that there are no torsion points of order n in C(K) if 3 ≤ n ≤ 2g. So, it is natural to study torsion points of order 2g + 1 (notice that the number of such points in C(K) is always even). Recently, the authors proved that there are infinitely many (for a given g) mutually nonisomorphic pairs (C, O) such that C(K) contains at least four points of order 2g + 1. In the present paper we prove that (for a given g) there are at most finitely many (up to a isomorphism) pairs (C, O) such that C(K) contains at least six points of order 2g + 1.
GROUP ACTION ON GENUS 7 CURVES AND THEIR WEIERSTRASS POINTS
On Automorphism Groups of Genus 7, 2009
In this work, we generalize the theory of elliptic modular functions, to the case of genus 7. We investigate the equations of all algebraic curves of genus 7, their automorphism groups and their link to modern algebraic geometry and the theory of hyperelliptic curves. We discuss the cyclic covers of any curve of genus 7, the local structure of the moduli space at the corresponding Weierstrass points for each curve. We show that the largest finite group acting as the full automorphism group of a hyperelliptic curve of genus 7 has order 64 and we find its equation. We then, obtain all the 3g-3=18 hyperelliptic curves of genus 7 and their full automorphism groups. We discover that there are merely three other finite groups of order greater than 64 acting on some non-hyperelliptic curves of genus 7. We also obtain the equations of the non-hyperelliptic curves.
On the distribution of Weierstrass points on Gorenstein quintic curves
Journal of the Egyptian Mathematical Society, 2016
This paper is concerned with developing a technique to compute in a very precise way the distribution of Weierstrass points on the members of any 1-parameter family C a , a ∈ C , of Gorenstein quintic curves with respect to the dualizing sheaf K C a. The nicest feature of the procedure is that it gives a way to produce examples of existence of Weierstrass points with prescribed special gap sequences, by looking at plane curves or, more generally, to subcanonical curves embedded in some higher dimensional projective space.
On Weierstrass points of Hurwitz curves
Journal of Algebra, 2006
Let X be a compact Riemannn surface, or curve for short. Let g ≥ 2 be its genus, and G its automorphism group. Then |G| ≤ 84(g − 1), the well known Hurwitz bound. Curves attaining this bound are called Hurwitz curves and the corresponding groups are called Hurwitz groups. Recall that a finite group is a Hurwitz group if and only if it is generated by three elements of orders 2, 3 and 7 whose product is 1. The only Hurwitz curves of genus g < 14 are the famous Klein curve (of genus 3) and the MacBeath curve [Mb]of genus 7, see [Co]. Their automorphism groups G = L 2 (7) respectively G = L 2 (8) act transitively on their Weierstrass points. In this paper we prove that there are no other Hurwitz curves with this property, except possibly the Hurwitz curves of genus 14.
Torsion Points of order 2g+1 on odd degree hyperelliptic curves of genus g
arXiv (Cornell University), 2019
Let K be an algebraically closed field of characteristic different from 2, g a positive integer, f (x) ∈ K[x] a degree 2g + 1 monic polynomial without multiple roots, C f : y 2 = f (x) the corresponding genus g hyperelliptic curve over K, and J the Jacobian of C f. We identify C f with the image of its canonical embedding into J (the infinite point of C f goes to the zero of the group law on J). It is known [9] that if g ≥ 2, then C f (K) contains no points of orders lying between 3 and 2g. In this paper we study torsion points of order 2g + 1 on C f (K). Despite the striking difference between the cases of g = 1 and g ≥ 2, some of our results may be viewed as a generalization of well-known results about points of order 3 on elliptic curves. E.g., if p = 2g + 1 is a prime that coincides with char(K), then every odd degree genus g hyperelliptic curve contains at most two points of order p. If g is odd and f (x) has real coefficients, then there are at most two real points of order 2g + 1 on C f. If f (x) has rational coefficients and g ≤ 51, then there are at most two rational points of order 2g + 1 on C f. (However, there exist odd degree genus 52 hyperelliptic curves over Q that have at least four rational points of order 105.