Generalized weight properties of resultants and discriminants, and applications to projective enumerative geometry

Laurent Busé 1 Thomas Dedieu 2
1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , National and Kapodistrian University of Athens
Abstract : In a book dating back to 1862, Salmon stated a formula giving the first terms of the Taylor expansion of the discriminant of a plane algebraic curve, and from it derived various enumerative quantities for surfaces in the 3-dimensional projective space. In this text, we provide complete proofs of this formula and its enumerative applications, and extend Salmon's considerations to hypersurfaces in a projective space of arbitrary dimension. To this end, we extend reduced elimination theory by introducing the concept of reduced discriminant, and provide a thorough study of its weight properties; the latter are deeply linked to projective enumerative geometric properties. Then, following Salmon's approach, we compute the number of members of a pencil of hyperplanes that are bitangent to a fixed projective hypersurface. Some other results in the same spirit are also discussed.
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Laurent Busé, Thomas Dedieu. Generalized weight properties of resultants and discriminants, and applications to projective enumerative geometry. 2018. ⟨hal-01936025⟩

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