Busé, Laurent; D’Andrea, Carlos; Sombra, Martín; Weimann, Martin The geometry of the flex locus of a hypersurface. (English) Zbl 1443.14045 Pac. J. Math. 304, No. 2, 419-437 (2020). 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. Reviewer: Hans-Peter Schröcker (Innsbruck) MSC: 14J70 Hypersurfaces and algebraic geometry 13P15 Solving polynomial systems; resultants 14N05 Projective techniques in algebraic geometry Keywords:hypersurface; flex point; flex locus; multivariate resultant PDFBibTeX XMLCite \textit{L. Busé} et al., Pac. J. Math. 304, No. 2, 419--437 (2020; Zbl 1443.14045) Full Text: DOI arXiv Link References: [1] 10.1007/978-3-0348-5097-1 · doi:10.1007/978-3-0348-5097-1 [2] ; Cox, Using algebraic geometry. Graduate Texts in Mathematics, 185 (2005) · Zbl 1079.13017 [3] 10.1017/CBO9781139062046 · Zbl 1341.14001 · doi:10.1017/CBO9781139062046 [4] 10.1007/978-0-8176-4771-1 · doi:10.1007/978-0-8176-4771-1 [5] 10.4007/annals.2015.181.1.2 · Zbl 1310.52019 · doi:10.4007/annals.2015.181.1.2 [6] 10.1353/ajm.2018.0028 · Zbl 1403.51002 · doi:10.1353/ajm.2018.0028 [7] 10.1007/s10208-002-0078-2 · Zbl 1058.14075 · doi:10.1007/s10208-002-0078-2 [8] ; Jouanolou, Théorèmes de Bertini et applications. Progress in Mathematics, 42 (1983) · Zbl 0519.14002 [9] 10.1016/0001-8708(91)90031-2 · Zbl 0747.13007 · doi:10.1016/0001-8708(91)90031-2 [10] ; Katz, Proceedings of the International Congress of Mathematicians, 303 (2014) [11] 10.1016/j.aim.2014.11.014 · Zbl 1309.14042 · doi:10.1016/j.aim.2014.11.014 [12] 10.1515/crll.1999.030 · Zbl 1067.14514 · doi:10.1515/crll.1999.030 [13] 10.1007/s00454-017-9940-5 · Zbl 1388.14109 · doi:10.1007/s00454-017-9940-5 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.