We show how the theory of multiplier ideals can be developed and discuss several applications of this theory. In the second section the same theory in the analytic setting is developed and several...Show moreWe show how the theory of multiplier ideals can be developed and discuss several applications of this theory. In the second section the same theory in the analytic setting is developed and several applications are given. Let X be a smooth algebraic variety and D an effective Q-divisor. We associate to D (or to the pair (X, D)) an ideal sheaf I(D) which controls the behavior of the fractional part of D and determines how close it is to have an simple normal crossing support. Other applications can be treated such as singularities of projective hypersurfaces and characterization of divisors. In the former case a result of Esnault-Viehweg concerning the least degree of hypersurfaces with multiplicity greater than or equal to a given positive integer at each point of a finite set is explained and proved in two different ways. A slight generalization is also given. Several vanishing and non-vanishing results including a global generation theorem are treated which will be used to prove the results about singularities. In the second section the analytic analogues of the materials in section one are given and the characterization of analytic nef and good divisors are explained.Show less
In this Bachelor Thesis, we will explain a calculus named Schubert Calculus. Schubert Calculus is invented by Hermann C¨asar Hannibal Schubert around the end of the nineteenth century. This...Show moreIn this Bachelor Thesis, we will explain a calculus named Schubert Calculus. Schubert Calculus is invented by Hermann C¨asar Hannibal Schubert around the end of the nineteenth century. This calculus allowed Schubert and his successors to solve many enumerative problems in geometry, although they didn’t have rigorous proofs of the rules in this calculus. This is the reason why Hilbert’s 15-th problem concerns with this calculus, and nowadays most of the rules in this calculus are finally formalized (through topology and intersection theory). The main purpose of this Bachelor Thesis is to explain the rules of this Schubert Calculus and solve some enumerative problems. The first chapter introduces the Grassmann Variety (mainly from [KL]), and the second chapter gives some basic facts about the cohomology ring of this Grassmann Variety (mainly based on [KL], [FU] and [ST]). In the third and the fifth chapter we will develop the calculus in this cohomology ring (mainly from [KL] and [ST]). The fourth chapter shows the power of the Schubert Calculus by solving several enumerative problems (many of which are new). I have decided not to include complete proofs of the formulae from the second chapter, since the complete proofs I know are very technical (although we will give a sketch). Proofs can be found, for example, in [GH] (although it contains some errors), [FU] (as exercises) and [HP] (but this is hard to read). For more details and proofs of Chapter Five, I suggest to read [FU]. I have also decided not to include (part of) the theory of Schubert Polynomials and Varieties, which is a current area of research, since a detailed introduction can be found in [FU].Show less
