Configuration spaces of algebraic varieties

Dette speciale omhandler en computerimplementering af en spektralfølge, som er beskrevet af Totaro, der kan bruges til udregning af den rationelle kohomologi af et konfigurationsrum $F(X,n)$ af $n$-tupler af punkter i en glat kompleks projektiv kurve $X$ af genus $g$. Første kapitel introducerer nogle teoretiske forkundskaber for implementeringen. Første afsnit introducerer nogle grundlæggende algebraiske strukturer. Afsnit 2 introducerer kohomologi af topologiske rum. Afsnit 3 omhandler orienterbare mangfoldigheder og kohomologiske egenskaber ved disse. Afsnit 4 introducerer spektralfølger og afsnit 5 omhandler komplekse algebraiske varieteter. Afsnit 6 opsummerer kort resultaterne af Totaros artikel. Kapitel 2 omhandler detaljerne ved implementeringen af computerprogrammet. Afsnit 1 indeholder detaljer omkring implementeringen af de underliggende algebraer. Afsnit 2 omhandler detaljer ved elementoperationer i disse algebraer of afsnit 3 omhandler implementeringen af differentialet for spektralfølgen, samt udregningen af kohomologien. Appendiks A indeholder et par eksempler på resultater fra programmet.
This thesis deals with a computer implementation of a spectral sequence by Totaro that can be used for the calculation of the rational cohomology of the configuration space $F(X,n)$ of ordered $n$-tuples of points in a smooth complex projective curve $X$ of genus $g$. The first chapter of the thesis introduces some theoretical prerequisites for the implementation. Section 1 introduces some basic algebraic objects used in the thesis. Section 2 introduces cohomology of spaces and the added algebraic structure coming from the cup product. Section 3 introduces manifolds and some cohomological properties of orientable manifolds. Section 4 briefly introduces spectral sequnces and section 5 deals with complex algebraic varieties, and finally section 6 briefly sums up the results of Totaro's paper. Chapter 2 contains the implementation details of the program. Section 1 contains details for the bases of the underlying algebras of the $E_2$-page of Totaro's spectral sequence. Section 2 contains details for operations with elements in these algebras and section 3 contains details for the differential of the spectral sequence and calculating cohomology of the algebra. Appendix A contains a few example outputs from the implementation.