Curves in cages: an algebro-geometric zoo.(English)Zbl 1156.14019

The author defines a $$(d \times e)$$-cage as the intersection of two unions $$\mathcal D$$ and $$\mathcal E$$ of $$d$$ (resp. $$e$$) lines in the real or complex projective plane. The lines in $$\mathcal D$$ are called “red”, the lines in $$\mathcal E$$ are called “blue”. The author studies algebraic curves of degree $$k$$ through “many” nodes of the cage.
A subset $$\mathcal A$$ of the cage nodes is called quasi-triangular if every blue line contains between $$d-e+1$$ and $$d$$ elements of $$\mathcal A$$ and each value is attained exactly once; it is called supra-quasi-triangular, if the node number is between $$d-e+2$$ and $$d$$ and each value between $$d-e+2$$ and $$d-1$$ is attained exactly once (then the number $$d$$ is attained twice). The following main results hold true:
1. “If a curve in $$\mathbb{P}^2$$ of degree $$d$$ passes through a supra-quasi-triangular set $$\mathcal A$$ of nodes of a $$(d \times e)$$-cage with $$d \geq e$$, then it passes through all nodes of the cage.”
2. “No curve of degree less then $$e$$ can pass through a quasi-triangular set of nodes of a $$(d \times e)$$-cage when $$d \geq e$$.”
3. Consider a $$(d \times d)$$-cage $$K$$, let $$R$$ and $$B$$ be the products of linear polynomials that describe its red and blue lines, respectively, and denote by $$S_{[\lambda:\mu]}$$ the curve defined by $$\lambda R + \mu B$$. Then every direction $$t$$ at one of the cage nodes $$p$$ defines a unique curve $$S_{[\lambda:\mu]}$$ that is tangent to $$t$$ in $$p$$. Moreover, any curve of degree $$d$$ through a supra-quasi-triangular subset $$\mathcal A$$ of $$K$$ having $$t$$ as tangent direction at $$p \in \mathcal A$$ contains all nodes of $$K$$.
4. If a polygon $$\mathcal D$$ with $$2d$$ sides, alternately colored in red and blue, is inscribed into a quadratic curve $$\mathcal Q$$ it generates a cage $$K$$ such that the $$d^2-d$$ nodes of $$K \setminus \mathcal Q$$ lie on a curve $$\mathcal Q_\star$$ of degree $$d-2$$ or less. Generically, $$\mathcal Q_\star$$ is unique.
Throughout the text the author highlights possible generalisations and relations to classical results such as the Theorem of Pascal on hexagons inscribed into a quadratic curve and Theorem of Chasles on complete intersections of cubic curves. Moreover, he pays attention to an elementary and entertaining treatment. Only in the concluding Section 4 he relates his findings to modern algebraic geometry by demonstrating how they can be derived from the Bacharach duality theorem.

MSC:

 14Hxx Curves in algebraic geometry 51N35 Questions of classical algebraic geometry
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