Related papers: Weierstrass weight of the hyperosculating points o…

Let $(S,H)$ be a generalized Fermat pair of the type $(k,n)$. If $F\subset S$ is the set of fixed points of the non-trivial elements of the group $H$, then $F$ is exactly the set of hyperosculating points of the standard embedding…

In this paper we determine the generalized Weierstrass semigroup $ \widehat{H}(P_{\infty}, P_1, \ldots , P_{m})$, and consequently the Weierstrass semigroup $H(P_{\infty}, P_1, \ldots , P_{m})$, at $m+1$ points on the curves…

In this paper we consider the problem of calculating the higher-order Weierstrass weight of the branch points of a superelliptic curve $C$. For any $q>1$, we give an exact formula for the $q$-weight of an affine branch point. We also find a…

A closed Riemann surface $S$ is called a generalized Fermat curve of type $(p,n)$, where $n,p \geq 2$ are integers such that $(p-1)(n-1)>2$, if it admits a group $H \cong {\mathbb Z}_{p}^{n}$ of conformal automorphisms with quotient…

Let $X$ denote an integral, projective Gorenstein curve over an algebraically closed field $k$. In the case when $k$ is of characteristic zero, C. Widland and the second author have defined Weierstrass points of a line bundle on $X$. In the…

In this paper we demonstrate that the notion of inflection points and extactic points on plane algebraic curves can be suitably transferred to curves in $\mathbb{P}^1\times \mathbb{P}^1$. More precisely, we describe osculating curves and…

We construct a general form for an F-theory Weierstrass model over a general base giving a 6D or 4D supergravity theory with gauge group $(\operatorname{SU}(3) \times \operatorname{SU}(2) \times \operatorname{U}(1)) / \mathbb{Z}_6$ and…

In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Giulietti-Korchm\'aros curve $\mathcal{X}$. We show that as the point varies, exactly three possibilities arise: One for the…

{\it $(N,\gamma)$-hyperelliptic} semigroups were introduced by Fernando Torres to encapsulate the most salient properties of Weierstrass semigroups associated to totally-ramified points of $N$-fold covers of curves of genus $\gamma$. Torres…

In this work we study the generalized Weierstrass semigroup $\widehat{H} (\mathbf{P}_m)$ at an $m$-tuple $\mathbf{P}_m = (P_{1}, \ldots , P_{m})$ of rational points on certain curves admitting a plane model of the form $f(y) = g(x)$ over…

A non-singular complete irreducible algebraic curve $F_{k,n}$, defined over an algebraically closed field $K$, is called a generalized Fermat curve of type $(k,n)$, where $n, k \geq 2$ are integers and $k$ is relatively prime to the…

In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Suzuki curve $\mathcal{S}_q$. As the point $P$ varies, exactly two possibilities arise for $H(P)$: one for the…

Given a hyperelliptic curve $C$ of genus $g$ over a number field $K$ and a Weierstrass model $\mathscr{C}$ of $C$ over the ring of integers ${\mathcal O}_K$ (i.e. the hyperelliptic involution of $C$ extends to $\mathscr{C}$ and the quotient…

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 of the hyperelliptic discriminant of X/S, and the valuation of the Mumford…

We determine the Weierstrass semigroup $H(P_\infty,P_1,\ldots,P_m)$ at several rational points on the maximal curves which cannot be covered by the Hermitian curve introduced by Tafazolian, Teher\'an-Herrera, and Torres. Furthermore, we…

In this work, we are concerned with the structure of sparse semigroups and some applications of them to Weierstrass points. We manage to describe, classify and find an upper bound for the genus of sparse semigroups. We also study the…

Let C be the union of two general connected, smooth, nonrational curves X and Y intersecting transversally at a point P. Assume that P is a general point of X or of Y. Our main result, in a simplified way, says: Let Q be a point of X. Then…

In this article we explicitly determine the Weierstrass semigroup at any place of some $\mathbb{F}_{q^2}$-maximal Fermat function fields $\mathcal{F}_m$, namely for $m=(q+1)/2$ and $m=(q+1)/3$. These famous function fields arise as Galois…

In this work, we investigate generalized Weierstrass semigroups in arbitrary Kummer extensions of function field $\mathbb{F}_q(x)$. We analyze their structure and properties, with a particular emphasis on their maximal elements. Explicit…

We determine the Weierstrass semigroup $H(P_{\infty}, P_{1}, \ldots , P_{m})$ at several points on the $GK$ curve. In addition, we present conditions to find pure gaps on the set of gaps $G(P_{\infty}, P_{1}, \ldots , P_{m})$. Finally, we…