Functors of Liftings of Projective Schemes

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 {\em...

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Bibliographic Details
Published in:arXiv.org 2018-07
Main Authors: Bertone, Cristina, Cioffi, Francesca, Franco, Davide
Format: Article
Language:English
Subjects:
Gluing
Hyperplanes
Inverse problems
Polynomials
Production methods
Online Access:Get full text
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Summary: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 {\em lifting problem}, and every such scheme \(W\) is called a {\em 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 Gr\"obner and marked bases theories. Furthermore, by classical tools we obtain an analogous result for equidimensional liftings. Examples of explicit computations are provided.
ISSN:2331-8422
DOI:10.48550/arxiv.1706.02618