Dissertation

## Computing Quot Schemes

##### Abstract

The main goal of this thesis is to develop computational methods which

allow effective computations on Hilbert and Quot schemes.

At first we introduce marked bases over modules. They may be considered as a

form of Gröbner basis which do not depend on a term order. Instead, one

chooses for each generator some term as head module term such that the head

module terms generate a prescribed monomial module. We show that the

involutive normal form algorithm with respect to Pommaret division will

terminate if the prescribed monomial module is quasi-stable. For the

introduction of marked bases over modules we introduce the concept of

resolving decompositions which provide a unifying framework for computing free

resolutions

Then we investigate into the Hilbert function and the Hilbert polynomial.

Furthermore, we analyse the important persistence and regularity theorem for

ideals of Gotzmann. We provide for both theorems new alternative proofs which

are much simpler to understand than previous proofs. They are based on the

theory of Pommaret bases.

We recall the definition of the Hilbert, Quot and Grassmann functors. Then we

construct the "quasi-stable covering" of the Grassmann functor and show that

we can restrict this covering to the Quot functor. The covering is represented

by subfunctors. In an additional step we show that every subfunctor can be

represented by a marked scheme. A marked scheme parametrizes all marked bases

which belong to a prescribed quasi-stable module.

Furthermore, we develop algorithms for the concrete computation of Quot schemes. At

first we investigate the computation of saturated quasi-stable or p-Borel

fixed monomial modules. At second we present two algorithms for computing

marked schemes. By using these algorithms we show that the Hilbert scheme of 4

points in the projective 3-space is reduced. Moreover, we compute for the

first time a concrete representation for a non-trivial Quot scheme and give an

example for Hilbert schemes over fields of finite characteristic.

allow effective computations on Hilbert and Quot schemes.

At first we introduce marked bases over modules. They may be considered as a

form of Gröbner basis which do not depend on a term order. Instead, one

chooses for each generator some term as head module term such that the head

module terms generate a prescribed monomial module. We show that the

involutive normal form algorithm with respect to Pommaret division will

terminate if the prescribed monomial module is quasi-stable. For the

introduction of marked bases over modules we introduce the concept of

resolving decompositions which provide a unifying framework for computing free

resolutions

Then we investigate into the Hilbert function and the Hilbert polynomial.

Furthermore, we analyse the important persistence and regularity theorem for

ideals of Gotzmann. We provide for both theorems new alternative proofs which

are much simpler to understand than previous proofs. They are based on the

theory of Pommaret bases.

We recall the definition of the Hilbert, Quot and Grassmann functors. Then we

construct the "quasi-stable covering" of the Grassmann functor and show that

we can restrict this covering to the Quot functor. The covering is represented

by subfunctors. In an additional step we show that every subfunctor can be

represented by a marked scheme. A marked scheme parametrizes all marked bases

which belong to a prescribed quasi-stable module.

Furthermore, we develop algorithms for the concrete computation of Quot schemes. At

first we investigate the computation of saturated quasi-stable or p-Borel

fixed monomial modules. At second we present two algorithms for computing

marked schemes. By using these algorithms we show that the Hilbert scheme of 4

points in the projective 3-space is reduced. Moreover, we compute for the

first time a concrete representation for a non-trivial Quot scheme and give an

example for Hilbert schemes over fields of finite characteristic.

##### Sponsorship

The author was partially supported by a fellowship by the Otto-Braun-Fonds##### Citation

@phdthesis{urn:nbn:de:hebis:34-2017022752117,

author={Albert, Mario},

title={Computing Quot Schemes},

school={Kassel, Universität Kassel, Fachbereich Mathematik und Naturwissenschaften, Institut für Mathematik},

month={02},

year={2017}

}

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2017-02-27T12:03:14Z 2017-02-27T12:03:14Z 2017-02-27 urn:nbn:de:hebis:34-2017022752117 http://hdl.handle.net/123456789/2017022752117 The author was partially supported by a fellowship by the Otto-Braun-Fonds eng Urheberrechtlich geschützt https://rightsstatements.org/page/InC/1.0/ Quot Schema Hilbert Schema Freie Auflösung Hilbert Polynom Markierte Basen Pommaret Basen Darstellbarkeit 510 Computing Quot Schemes Dissertation The main goal of this thesis is to develop computational methods which allow effective computations on Hilbert and Quot schemes. At first we introduce marked bases over modules. They may be considered as a form of Gröbner basis which do not depend on a term order. Instead, one chooses for each generator some term as head module term such that the head module terms generate a prescribed monomial module. We show that the involutive normal form algorithm with respect to Pommaret division will terminate if the prescribed monomial module is quasi-stable. For the introduction of marked bases over modules we introduce the concept of resolving decompositions which provide a unifying framework for computing free resolutions Then we investigate into the Hilbert function and the Hilbert polynomial. Furthermore, we analyse the important persistence and regularity theorem for ideals of Gotzmann. We provide for both theorems new alternative proofs which are much simpler to understand than previous proofs. They are based on the theory of Pommaret bases. We recall the definition of the Hilbert, Quot and Grassmann functors. Then we construct the "quasi-stable covering" of the Grassmann functor and show that we can restrict this covering to the Quot functor. The covering is represented by subfunctors. In an additional step we show that every subfunctor can be represented by a marked scheme. A marked scheme parametrizes all marked bases which belong to a prescribed quasi-stable module. Furthermore, we develop algorithms for the concrete computation of Quot schemes. At first we investigate the computation of saturated quasi-stable or p-Borel fixed monomial modules. At second we present two algorithms for computing marked schemes. By using these algorithms we show that the Hilbert scheme of 4 points in the projective 3-space is reduced. Moreover, we compute for the first time a concrete representation for a non-trivial Quot scheme and give an example for Hilbert schemes over fields of finite characteristic. open access Albert, Mario Kassel, Universität Kassel, Fachbereich Mathematik und Naturwissenschaften, Institut für Mathematik Seiler, Werner M. (Prof. Dr.) Roggero, Margherita (Prof.) 13P10 13D02 14C05 14Q20 Hilbertsches Schema Darstellbarkeit

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