In Chapter 1 we study the moduli spaces Simpmn of degree n + 1 morphisms A1K ! A1K with "ramification length < m" over an algebraically closed field K. For each m, the moduli space Simpmn is a Zariski open subset of the space of degree n + 1 polynomials over K up to Aut(A1K). It is, in a way, orthogonal to the many papers about polynomials with prescribed zeroes- here we are prescribing, instead, the ramification data. Exploit- ing the topological properties of the poset that encodes the ramification behaviour, we use a sheaf-theoretic argument to compute H⇤(Simpmn (C);Q) as well as the étale coho- mology H⇤(Simpm ;Q ) for charK = 0 or charK > n+1. As a by-product we obtain e ́t n/K` that H⇤(Simpmn (C); Q) is independent of n, thus implying rational cohomological stability. When charK > 0 our methods compute H⇤ (Simpm; Q ) provided charK > n + 1 and e ́t n ` show that the étale cohomology groups in positive characteristics do not stabilize. In Chapter 2, inspired by Deligne’s use of the simplicial theory of hypercoverings in defining mixed Hodge structures ([Del75]), we define the notion of semi-simplicial filtration of a family of spaces by some fixed space. A result of the semi-simplicial filtration is the existence of natural open subsets- the ‘unfiltered strata’ or the ‘zeroth strata’. In this paper, for a family of spaces {Xn} admitting a semi-simplicial filtration with zeroth strata {Un}, we construct a spectral sequence, somewhat like the Czech-to-derived category spectral sequence. Instances where our methods can be applied to retrieve old results include computation of (in some examples, stable) Q-Betti numbers of the moduli space of basepoint free pencils over a smooth projective curve, moduli space of monic degree n square-free polynomials etc.




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