Computing Quot schemes via marked bases over quasi-stable modules

Albert, Mario; Bertone, Cristina; Roggero, Margherita; Seiler, Werner M.

doi:10.48550/arxiv.1511.03547

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“…The assumption that J ′ is quasi-stable is not restrictive: indeed, this can be obtained by a change of coordinates on I ′ , and this change does not effect the scheme Y = Proj (K[x]/I ′ ) from a geometric point of view. Quasi-stability for initial ideals will allow us to use the techniques concerning marked bases over quasi-stable ideals developed in (Bertone et al, 2017) (see (Albert et al, 2016) for the more general case of free modules).…”

Section: The Case Of a Quasi-stable Initial Idealmentioning

confidence: 99%

Functors of liftings of projective schemes

Bertone

Cioffi

Franco

2019

Journal of Symbolic Computation

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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.

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Section: The Case Of a Quasi-stable Initial Idealmentioning

confidence: 99%