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Resolutions of singularities using blow-ups
Komora, Matúš ; Šťovíček, Jan (advisor) ; Hrbek, Michal (referee)
This bachelor's thesis aims to provide accessible treatment of the blow-up construction for algebraic varieties. The blow-up construction is a fundamen- tal technique in algebraic geometry that allows us to find a variety which has better properties than an original variety but is still equivalent to the original. This process can be used to resolve singularities. In the first two chapters, we begin by providing an introduction to the fundamental principles of alge- braic geometry, including the definitions of algebraic varieties but also basic topological concepts but also some construction such as Segre embedding and product of varieties. In the third chapter, we will introduce the concept of blow-ups and show the computation as on example. 1
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Toric varieties and their applications
Klepáč, Adam ; Šťovíček, Jan (advisor) ; Williamson, Jordan (referee)
The thesis provides an introduction into the theory of affine and abstract toric vari- eties. In the first chapter, tools from algebraic geometry indispensable for the compre- hension of the topic are introduced. Many properties of convex polyhedral cones and affine toric varieties are proven and discussed in detail as is the deep connection between the two objects. The second chapter establishes the notion of an abstract variety and translates obtained results to this more general setting, giving birth to the theory of abstract toric varieties and the closely associated theory of fans. Finally, an algorithmic approach to the resolution of singularities on toric surfaces and its relation to continued fractions is revealed. 1
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Module approximations and direct limits
Matoušek, Cyril ; Šaroch, Jan (advisor) ; Šťovíček, Jan (referee)
This master's thesis deals with questions about the existence of module appro- ximations, namely C-precovers and C-covers for a given class C of R-modules, and studies the relations of these approximations with direct limits. Thanks to a the- orem due to Enochs, we know that every R-module has a C-cover if the pre- covering class C is closed under direct limits, although the validity of the con- verse implication remains an open problem known as Enochs' conjecture. In this setting, we show that any module M with perfect decomposition satisfies that the class Add(M) is precovering and closed under direct limits; hence also cove- ring. Furthermore, we prove Enochs' conjecture for Add(M) if M is small, e.g. < ℵω-generated. Specifically, if M is small and Add(M) covering, then M has a perfect decomposition.
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Vychylující teorie pro kvazikoherentní svazky
Čoupek, Pavel ; Šťovíček, Jan (advisor) ; Trlifaj, Jan (referee)
We introduce the definition of 1-cotilting object in a Grothendieck category and investigate its relation to the analogue of the standard definition of 1-cotilting module. The 1-cotilting quasi-coherent sheaves on a Noetherian scheme are stud- ied in particular: using the classification of hereditary torsion pairs in the category of quasi-coherent sheaves on a Noetherian scheme X, to each hereditary torsion- free class F that is generating we assign a 1-cotilting quasi-coherent sheaf whose 1-cotilting class is F. This provides a family of pairwise non-equivalent 1-cotilting quasi-coherent sheaves which are parametrized by specialization closed subsets of X avoiding the set of associated points of a chosen generator of the category of quasi-coherent sheaves. In many cases (e.g. for separated schemes), this set of avoided points can be chosen as the set of associated points of the scheme. 1
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Samoopravné kódy a rozpoznávání podle duhovky
Luhan, Vojtěch ; Šťovíček, Jan (advisor) ; Drápal, Aleš (referee)
Iris recognition constitutes one of the most powerful method for the iden- tification and authentication of people today. This thesis aims to describe the algorithms used in a sophisticated and mathematically correct way, while re- maining comprehensible. The description of these algorithms is not the only objective of this thesis; the reason they were chosen and potential improvements or substitutions are also discussed. The background of iris recognition, its use in cryptosystems, and the application of error-correcting codes are investigated as well.
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Point Counting on Elliptic and Hyperelliptic Curves
Vácha, Petr ; Šťovíček, Jan (advisor) ; Drápal, Aleš (referee)
In present work we study the algorithms for point counting on elliptic and hy- perelliptic curves. At the beginning we describe a few simple and ineffective al- gorithms. Then we introduce more complex and effective ways to determine the point count. These algorithms(especially the Schoof's algorithm) are important for the cryptography based on discrete logarithm in the group of points of an el- liptic or hyperelliptic curve. The point count is important to avoid the undesirable cases where the cryptosystem is easy to attack. 1
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