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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]
A geometric solution to this is to intersect the curve C not with itself, but with a slightly pushed off version of itself. In the plane, this just means translating the curve C in some direction, but in general one talks about taking a curve C′ that is linearly equivalent to C , and counting the intersection C · C′ , thus obtaining an ...
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.
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:
666841088 The number of quadric surfaces tangent to 9 given quadric surfaces in general position in 3-space (Schubert 1879, p.106) (Fulton 1984, p. 193) 5819539783680 The number of twisted cubic curves tangent to 12 given quadric surfaces in general position in 3-space (Schubert 1879, p.184) (S. Kleiman, S. A. Strømme & S. Xambó 1987)
A smooth connected curve over an algebraically closed field is called hyperbolic if + > where g is the genus of the smooth completion and r is the number of added points.. Over an algebraically closed field of characteristic 0, the fundamental group of X is free with + generators if r>0.
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