Computing Quot Schemes

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 theore is 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.