Paul Aleksander Maugesten

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.

math.AG↗

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.

math.AG↗