Functors of liftings of projective schemes. (September 2019)

• Record Type: Journal Article
• Authors: Bertone, Cristina; Cioffi, Francesca; Franco, Davide
• Is Part Of: Journal of symbolic computation. Volume 94(2019)
• Page Start: 105
• Page End: 125
• Publication Date: 2019-09
• Subjects: Lifting -- Gröbner basis -- Marked basis -- Equidimensionality; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2018.07.003
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: A classical approach to investigate a closed projective scheme W consists of considering a general hyperplane section of W, which inherits many properties of W . The inverse problem that consists in finding a scheme W starting from a possible hyperplane section Y is called a lifting problem, and every such scheme W is called a lifting of Y . Investigations in this topic can produce methods to obtain schemes with specific properties. For example, any smooth point for Y is smooth also for W . We characterize all the liftings of Y with a given Hilbert polynomial by a parameter scheme that is obtained by gluing suitable affine open subschemes in a Hilbert scheme and is described through the functor it represents. We use constructive methods from Gröbner and marked bases theories. Furthermore, by classical tools we obtain an analogous result for equidimensional liftings. Examples of explicit computations are provided.