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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 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.)
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by Claude Chevalley ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles ) in a similar way to how simplicial or cellular homology ...
The twisted cubic is a projective algebraic variety. Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in ...
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} .
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)
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.
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:
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