Any cover of the Riemann sphere with rational branch points is known to be defined over the algebraic numbers. Hence the Galois group of the rationals acts on the category of such branched covers. Particulars about this action are still scarce, even in the simplest non-abelian case, the case with just three branch points. The first paper in this thesis describes a new algorithm, which uses modular form techniques in order to compute the equations for a cover of the Riemann sphere which is hyperelliptic as a curve. Given such equations one may easily determine the Galois orbit to which the cover belongs. We compute and discuss all covers of degree 6 and genus 2, and complete the case of covers of degree 7 and genus 1 as well. The second paper gives a proof of a formula for the number of three-point G-covers with a fixed special G-deformation datum (here G is a finite group which is strictly divisible by a prime number p). Since such a datum is an invariant for the action of the inertia group at p, this gives partial information about the action of this inertia group.

This thesis concerns computation and algebraic geometry. On the computational side we have focused on numerical homotopy methods. These procedures may be used to numerically solve systems of polynomial equations. The thesis contains four papers. In Paper I and Paper II we apply continuation techniques, as well as symbolic algorithms, to formulate methods to compute Chern classes of smooth algebraic varieties. More specifically, in Paper I we give an algorithm to compute the degrees of the Chern classes of smooth projective varieties and in Paper II we extend these ideas to cover also the degrees of intersections of Chern classes. In Paper III we formulate a numerical homotopy to compute the intersection of two complementary dimensional subvarieties of a smooth quadric hypersurface in projective space. If the two subvarieties intersect transversely, then the number of homotopy paths is optimal. As an application we give a new solution to the inverse kinematics problem of a six-revolute serial-link mechanism. Paper IV is a study of curves on certain special quartic surfaces in projective 3-space. The surfaces are invariant under the action of a finite group called the level (2,2) Heisenberg group. In the paper, we determine the Picard group of a very general member of this family of quartics. We have found that the general Heisenberg invariant quartic contains 320 smooth conics and we prove that in the very general case, this collection of conics generates the Picard group. / QC 20101115

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This thesis consists of a summary and three articles. The thesis is devoted to the study of knots and surfaces with additional geometric structures compared to the classical smooth structure. In Paper I, real algebraic rational knots in real projective space are studied up to rigid isotopy and we show that two real rational algebraic knots of degree at most 5 are rigidly isotopic if, and only if, their degree and encomplexed writhe are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree at most 6. Furthermore, an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented. In Paper II, we construct an invariant of parametrized generic real algebraic surfaces in real projective space which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: an intersection of two real sheets, an intersection of two complex conjugate sheets or a Whitney umbrella. The Brown invariant was expressed through a self linking number of the self intersection by Kirby and Melvin. We extend their definition of this self linking number to the case of parametrized generic real algebraic surfaces. In Paper III, we give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in the product of a punctured Riemann surface with the real line. As an application we show that for any nonzero homology class h, and for any integer k there exist k Legendrian knots all representing h which are pairwise smoothly isotopic through a formal Legendrian isotopy but which lie in mutually distinct Legendrian isotopy classes.