arXiv subjects

Torgunn Karoline Moe

Publications and source records attributed to Torgunn Karoline Moe.

8 recordsLinked to original sources

arXiv2025

The Fermat curves, arrangements of lines, and intersections of osculating curves

In this paper we present new results about arrangements of lines and osculating curves associated to the Fermat curves in the projective plane. We first consider the sextactic points on the Fermat curves and show that they are distributed on three grids. The grid lines constitute new line arrangements and examples of free curves associated with the Fermat curves. Moreover, we compute the hyperosculating conics to the Fermat curves, study the arrangement of these conics, and find that they intersect in a special way. The latter result is a consequence of the action of the group of automorphisms on osculating curves, and we conclude with a more general result for intersections of osculating curves of any given degree.

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arXiv2018

The 2-Hessian and sextactic points on plane algebraic curves

In an article from 1865, Arthur Cayley claims that given a plane algebraic curve there exists an associated 2-Hessian curve that intersects it in its sextactic points. In this paper we fix an error in Cayley's calculations and provide the correct defining polynomial for the 2-Hessian. In addition, we present a formula for the number of sextactic points on cuspidal curves and tie this formula to the 2-Hessian. Lastly, we consider the special case of rational curves, where the sextactic points appear as zeros of the Wronski determinant of the 2nd Veronese embedding of the curve.

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arXiv2018

Special Weierstrass points on algebraic curves in $\mathbb{P}^1\times \mathbb{P}^1$

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 study Weierstrass points of algebraic curves in the surface $\mathbb{P}^1\times \mathbb{P}^1$ with respect to certain linear systems. In particular, we study points where a fiber of $\mathbb{P}^1\times \mathbb{P}^1$ is tangent, and points with a hyperosculating $(1,1)$-curve. In the first case we find Hessian-like curves that intersect the curve in these points, and in the second case we find a local criteria. Moreover, we provide Plücker-like formulas for the number of smooth Weierstrass points on a curve. In the special case of rational curves, we use suitable Wronskians to compute these points and their respective Weierstrass weights.

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arXiv2015

Rational Cuspidal Curves

Submission on request. This Master thesis from 2008 (University of Oslo, Norway) contains no new results, but it provides an overview of plane rational cuspidal curves, in particular curves of low degree. Please note that new results on this topic and the related topic of rational cuspidal curves on Hirzebruch surfaces have been published after this thesis was written.

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arXiv2014

Topological obstructions for rational cuspidal curves in Hirzebruch surfaces

We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer theory and is a generalization of a result by Livingston and the first author. The second criterion is obtained by comparing the spectrum of a suitably defined link at infinity of a curve with spectra of its singular points.

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arXiv2013

On the number of cusps on cuspidal curves on Hirzebruch surfaces

In this article we give an upper bound for the number of cusps on a cuspidal curve on a Hirzebruch surface. We adapt the results that have been found for a similar question asked for cuspidal curves on the projective plane, and restate the results in this new setting.

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arXiv2013

Rational cuspidal curves with four cusps on Hirzebruch surfaces

The purpose of this article is to shed light on the question of how many and what kind of cusps a rational cuspidal curve on a Hirzebruch surface can have. We use birational transformations to construct rational cuspidal curves with four cusps on the Hirzebruch surfaces and find associated results for these curves.

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arXiv2013

Segre Classes on Smooth Projective Toric Varieties

We provide a generalization of the algorithm of Eklund-Jost-Peterson for computing Segre classes of closed subschemes of projective k-space. The algorithm is here generalized to computing the Segre classes of closed subschemes of smooth projective toric varieties.

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