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The geometry of the flex locus of a hypersurface. (English) Zbl 1443.14045

Generalizing the notion of inflexion points of a curve, a point \(p\) of a projective hypersurface is called flex point if a straight line exists that intersects the hypersurface in \(p\) with unexpectedly high order of contact. Flex points on curves in \(\mathbb{P}^2\) and surfaces in \(\mathbb{P}^3\) were a topic of 19th century mathematics This paper investigates flex points in arbitrary dimension. Since the flex locus for ruled surfaces are trivial and algebraic surfaces of degree \(d < n\) in \(\mathbb{P}^n\) are ruled, one naturally assumes \(d \ge n\).
A first result proves the existence of a unique (modulo the hypersurface equation) polynomial for the equation of the flex locus in terms of multivariate resultants. It allows to bound the flex locus’ degree and generalizes classical results in dimension two and three. An open problem is to find a canonical representative. For generic hypersurfaces, the previously obtained bound is shown to be sharp. Moreover, flex lines for generic flex points are unique. These results seem to be new even in case of \(n \in \{2,3\}\).
Flex loci have recently been recognized as important objects in incidence geometry. Researchers from that area will certainly appreciate the obtained results and also the formulation of the theory in the language of modern algebraic geometry.

MSC:

14J70 Hypersurfaces and algebraic geometry
13P15 Solving polynomial systems; resultants
14N05 Projective techniques in algebraic geometry
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