Rational Points on Algebraic Varieties and Geometry of Curves and Surfaces

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This project addresses several fundamental questions inalgebraic geometry and number theory.The first part concerns the geometry of solutions toalgebraic equations.Let X be a variety defined over a number field K.Does there exist a finite extension L/K such thatX(L) is dense? One seeks criteria for when thisoccurs in terms of the geometry of X.It is still not clear what form such criteria should take. The second part concerns the geometry of plane curves.When is an abstract curve a degenerationof smooth plane curves? What geometric properties dosuch limiting plane curves have in common?The third part involves the classification of surfacesingularities. Given a family of projective normalsurfaces, when are the invariants of such a familyconstant? Specifically, how does one tell whetherthe self-intersection of the canonical divisor is constant? The last part involves finding explicitparametrizations for the curves of a fixed small genus.Algebraic geometers study the structure of solutions topolynomial equations. From ancient times, geometricfigures like circles and ellipses have fascinated architects, scientists, and artists. The most compactway to represent such a figure is to describe it as the solutions to an equation. One basic question is to classify the abstract figures that may be representedby equations of a given form. Another is to understandthe common geometric properties of the solutions toall the equations of a given form. This project seeksto answer these questions for special types of curves and surfaces.