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

TL;DR

This paper extends the concept of inflection and Weierstrass points from plane algebraic curves to curves in the product surface  , providing formulas, criteria, and computational methods for these special points.

Contribution

It introduces a transfer of inflection point theory to   surfaces, including new formulas, local criteria, and computational techniques for Weierstrass points.

Findings

Derived Hessian-like curves for tangent points.

Established local criteria for hyperosculating points.

Provided Plfccker-like formulas for counting Weierstrass points.

Abstract

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 Pl\"ucker-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.