First order deformations of cones over projectively normal hyperelliptic curves

Let X be a smooth, proper algebraic variety over an algebraically ciosea field k, and let L be an ample, priojectively normal invertible sheaf in X. ..

The Weierstrass semigroups on double covers of genus two curves

We show that three numerical semigroups , and are of double covering type, i.e., the Weierstrass semigroups of ramification points on double covers of curves. Combining this with the results of Oliveira-Pimentel and Komeda we can determine the Weierstrass semigroups of the ramification points on double covers of genus two curves.Comment: 5 page

Weierstrass Points with First Non-Gap Four on a Double Covering of a Hyperelliptic Curve

2000 Mathematics Subject Classification: Primary 14H55; Secondary 14H30, 14H40, 20M14.Let H be a 4-semigroup, i.e., a numerical semigroup whose minimum positive element is four. We denote by 4r(H) + 2 the minimum element of H which is congruent to 2 modulo 4. If the genus g of H is larger than 3r(H) − 1, then there is a cyclic covering π : C −→ P^1 of curves with degree 4 and its ramification point P such that the Weierstrass semigroup H(P) of P is H (Komeda [1]). In this paper it is showed that we can construct a double covering of a hyperelliptic curve and its ramification point P such that H(P) is equal to H even if g ≤ 3r(H) − 1.* Partially supported by Grant-in-Aid for Scientific Research (15540051), Japan Society for the Promotion of Science. ** Partially supported by Grant-in-Aid for Scientific Research (15540035), Japan Society for the Promotion of Science

Corrigendum for "Weierstrass Points with first Non-Gap four on a Double Covering of a Hyperelliptic Curve"

In the proof of Lemma 3.1 in [1] we need to show that we may take the two points p and q with p ≠ q such that p+q+(b-2)g21(C′)∼2(q1+… +qb-1) where q1,…,qb-1 are points of C′, but in the paper [1] we did not show that p ≠ q. Moreover, we hadn't been able to prove this using the method of our paper [1]. So we must add some more assumption to Lemma 3.1 and rewrite the statements of our paper after Lemma 3.1. The following is the correct version of Lemma 3.1 in [1] with its proof

Weierstrass Points with First Non-Gap Four on a Double Covering of a Hyperelliptic Curve II

2000 Mathematics Subject Classification: Primary 14H55; Secondary 14H30, 14J26.A 4-semigroup means a numerical semigroup whose minimum positive integer is 4. In [7] we showed that a 4-semigroup with some conditions is the Weierstrass semigroup of a ramification point on a double covering of a hyperelliptic curve. In this paper we prove that the above statement holds for every 4-semigroup

On γ-Hyperelliptic Weierstrass Semigroups of Genus 6γ + 1 and 6γ

Let (C, P) be a pointed non-singular curve such that the Weierstrass semigroup H(P) of P is a γ-hyperelliptic numerical semigroup. Torres showed that there exists a double covering π : C → C‘ such that the point P is a ramification point of π if the genus g of C is larger than or equal to 6γ + 4. Kato and the authors also showed that the same result holds in the case g = 6γ + 3 or 6γ + 2. In this paper we prove that there exists a double covering π : C → C’ satisfying the above condition even if g = 6γ + 1, 6γ and H(P) does not contain 4