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.


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