• The Pure and Applied Mathematics
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• v.22 no.4
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• pp.403-412
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• 2015

Kato[6] and Torres[9] characterized the Weierstrass semigroup of ramification points on h-hyperelliptic curves. Also they showed the converse results that if the Weierstrass semigroup of a point P on a curve C satisfies certain numerical condition then C can be a double cover of some curve and P is a ramification point of that double covering map. In this paper we expand their results on the Weierstrass semigroup of a ramification point of a double covering map to the Weierstrass semigroup of a pair (P, Q). We characterized the Weierstrass semigroup of a pair (P, Q) which lie on the same fiber of a double covering map to a curve with relatively small genus. Also we proved the converse: if the Weierstrass semigroup of a pair (P, Q) satisfies certain numerical condition then C can be a double cover of some curve and P, Q map to the same point under that double covering map.

• Journal of the Korean Mathematical Society
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• v.56 no.6
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• pp.1463-1474
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• 2019

Let C be a nonsingular projective curve of genus ${\geq}2$ over an algebraically closed field of characteristic 0. For a point P in C, the Weierstrass semigroup H(P) is defined as the set of non-negative integers n for which there exists a rational function f on C such that the order of the pole of f at P is equal to n, and f is regular away from P. A point P in C is referred to as a weak Galois-Weierstrass point if P is a Weierstrass point and there exists a Galois morphism ${\varphi}:C{\rightarrow}{\mathbb{p}}^1$ such that P is a total ramification point of ${\varphi}$. In this paper, we investigate the number of weak Galois-Weierstrass points of which the Weierstrass semigroups are generated by two positive integers.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.695-698
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• 2008

A simple proof of the fact that the j-invariants for Weierstrass equations are invariant under birational transformations which keep the forms of Weierstrass equations is given by finding a non-trivial explicit birational transformation which sends a normalized Weierstrass equation to the same equation.

• The Pure and Applied Mathematics
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• v.27 no.4
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• pp.251-267
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• 2020

We classify all semigroups each of which arises as a Weierstrass semigroup at a pair of non-Weierstrass points on a smooth plane curve of degree 5. First we find the candidates of semigroups by computing the dimensions of linear series on the curve. Then, by constructing examples of smooth plane curves of degree 5, we prove that each of the candidates is actually a Weierstrass semigroup at some pair of points on the curve. We need to study the systems of quadratic curves, which cut out the canonical series on the plane curve of degree 5.

• The Pure and Applied Mathematics
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• v.28 no.4
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• pp.399-411
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• 2021

We consider a Weierstrass semigroup at a generalized flex on a smooth plane curve. We find the candidates of a Weierstrass semigroup at a 2-flex of higher multiplicity on a smooth plane curve of degree d ≥ 5, and give some examples to show the existence of them.

• The Pure and Applied Mathematics
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• v.29 no.4
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• pp.353-368
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• 2022

In the previous paper [4], we classified the Weierstrass semigroups at a pair of inflection points of multiplicities d and d - 1 on a smooth plane curve of degree d. In this paper, as a continuation of those results, we classify all semigroups each of which arises as a Weierstrass semigroup at a pair of inflection points of multiplicities d, d - 1 and d - 2 on a smooth plane curve of degree d.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.4
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• pp.467-470
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• 2008

Abstract. We show that the cusp 1=2 is a Weierstrass point of ${\Gamma}_1(4p)$ if p is a prime greater than 7.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.379-384
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• 2015

In this paper, we find all Weierstrass points on the hyperelliptic modular curves $X_1(N)$.

• Journal of the Korean Mathematical Society
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• v.54 no.1
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• pp.69-86
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• 2017

The point $P{\in}{\mathbb{P}}^2$ is referred to as a Galois point for a nonsingular plane algebraic curve C if the projection ${\pi}_P:C{\rightarrow}{\mathbb{P}}^1$ from P is a Galois covering. In contrast, the point $P^{\prime}{\in}C^{\prime}$ is referred to as a weak Galois Weierstrass point of a nonsingular algebraic curve C' if P' is a Weierstrass point of C' and a total ramification point of some Galois covering $f:C^{\prime}{\rightarrow}{\mathbb{P}}^1$. In this paper, we discuss the following phenomena. For a nonsingular plane curve C with a Galois point P and a double covering ${\varphi}:C{\rightarrow}C^{\prime}$, if there exists a common ramification point of ${\pi}_P$ and ${\varphi}$, then there exists a weak Galois Weierstrass point $P^{\prime}{\in}C^{\prime}$ with its Weierstrass semigroup such that H(P') = or , which is a semigroup generated by two positive integers r and 2r + 1 or 2r - 1, such that P' is a branch point of ${\varphi}$. Conversely, for a weak Galois Weierstrass point $P^{\prime}{\in}C^{\prime}$ with H(P') = or , there exists a nonsingular plane curve C with a Galois point P and a double covering ${\varphi}:C{\rightarrow}C^{\prime}$ such that P' is a branch point of ${\varphi}$.

• Bulletin of the Korean Mathematical Society
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• v.44 no.2
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• pp.369-378
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• 2007

We classify all semigroups each of which arises as a Weierstrass semigroup at a pair of inflection points of multiplicities d or d-1 on a smooth plane curve of degree d.