Open AccessDissertation
Geometry of the projectivization of ideals and applications to problems of birationality
TLDR
In this article, the authors interpret geometrically the torsion of the symmetric algebra of the ideal sheaf I_Z of a scheme Z defined by n+1 equations in an n-dimensional variety.Abstract:
In this thesis, we interpret geometrically the torsion of the symmetric algebra of the ideal sheaf I_Z of a scheme Z defined by n+1 equations in an n-dimensional variety. This is equivalent to study the geometry of the projectivization of I_Z. The applications of this point of view concern, in particular, the topic of birational maps of the projective space of dimension 3 for which we construct explicit birational maps that have the same algebraic degree as their inverse, free and nearly-free curves for which we generalise a characterization of free curves by extending the notion of Milnor and Tjurina numbers. We tackle also the topic of homaloidal hypersurfaces, our original motivation, for which we produce in particular a homaloidal curve of degree 5 in characteristic 3. The last application concerns the computation of the inverse of a birational map.read more
References
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Book
Singular points of complex hypersurfaces
TL;DR: The Singular Points of Complex Hypersurfaces (AM-61) as mentioned in this paper is a seminal work in the area of complex hypersurfaces, and is based on as mentioned in this paper.
Book
Classical Algebraic Geometry: A Modern View
TL;DR: In this paper, the authors present a survey of the geometry of lines and cubic surfaces, including determinantal equations, theta characteristics, and the Cremona transformations.
Book
The geometry of schemes
David Eisenbud,Joe Harris +1 more
TL;DR: Projective Schemes and Local Constructions: Projective schemes and functors as mentioned in this paper are a generalization of classical constructions and local constructions, and they have been used in many applications.