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His Ph.D. thesis, written under the supervision of Gerard Washnitzer, was on The fundamental group of an algebraic curve. Fulton worked at Princeton and Brandeis University from 1965 until 1970, when he began teaching at Brown. In 1987 he moved to the University of Chicago. [1]
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimension at least 1. It is named after William Fulton and Johan Hansen, who proved it in 1979.
A key example of self-intersection numbers is the exceptional curve of a blow-up, which is a central operation in birational geometry. Given an algebraic surface S, blowing up at a point creates a curve C. This curve C is recognisable by its genus, which is 0, and its self-intersection number, which is −1. (This is not obvious.)
The characteristic linear system of a family of curves on an algebraic surface Y for a curve C in the family is a linear system formed by the curves in the family that are infinitely near C. [ 4 ] In modern terms, it is a subsystem of the linear system associated to the normal bundle to C ↪ Y {\displaystyle C\hookrightarrow Y} .
Let X be a Riemann surface.Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function :), we can associate a differential form of compact support, the Poincaré dual of c, with the property that integrals along c can be calculated by integrals over X:
Enumerative geometry saw spectacular development towards the end of the nineteenth century, at the hands of Hermann Schubert. [2] He introduced it for the purpose of Schubert calculus, which has proved of fundamental geometrical and topological value in broader areas.
The classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. fields that are not algebraically closed excluding the real numbers.
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