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Sparse versions of the Cayley-Bacharach theorem. (English) Zbl 1482.51019

The Cayley-Bacharach theorem is a well-known result from classical algebraic geometry. It states that all cubic curves through eight given points in the general position in the plane also pass through a unique ninth point. In this way, the Cayley-Bacharach theorem provides a map assigning a point in the plane to eight given points which is also called the extra point map. In [Q. Ren et al., Am. Math. Mon. 122, No. 9, 845–854 (2015; Zbl 1346.14083)], it is shown that the induced map is expressed in a rational form explicitly. In the paper under review, another formula for the induced map is proved, and a generalization of the Cayley-Bacharach theorem is obtained. The authors present this generalization as follows: “A choice of \(N\) generic points in \(\mathbb{C}^{d}\) gives rise to \(d\) hypersurfaces satisfying given constraints, and it turns out that those hypersurfaces meet in exactly \( N +1\) points.” Moreover, they prove that the corresponding generalized extra point map is rational. The paper concludes with a big amount of examples.

MSC:

51N35 Questions of classical algebraic geometry
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
13P15 Solving polynomial systems; resultants

Citations:

Zbl 1346.14083

Software:

Macaulay2
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Full Text: DOI arXiv

References:

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