Castelnuovo curve

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In algebraic geometry, a Castelnuovo curve, studied by Castelnuovo  ( 1889 ), is a curve in projective space Pn of maximal genus g among irreducible non-degenerate curves of given degree d.

Guido Castelnuovo Italian mathematician (1865–1952)

Guido Castelnuovo was an Italian mathematician. He is best known for his contributions to the field of algebraic geometry, though his contributions to the study of statistics and probability theory are also significant.

Castelnuovo showed that the maximal genus is given by the Castelnuovo bound

where m and ε are the quotient and remainder when dividing d–1 by n–1. Castelnuovo described the curves satisfying this bound, showing in particular that they lie on either a rational normal scroll or on the Veronese surface.

In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giuseppe Veronese (1854–1917). Its generalization to higher dimension is known as the Veronese variety.

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The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings.

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In the theory of algebraic curves, Brill–Noether theory, introduced by Alexander von Brill and Max Noether (1874), is the study of special divisors, certain divisors on a curve C that determine more compatible functions than would be predicted. In classical language, special divisors move on the curve in a "larger than expected" linear system of divisors.

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In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form.

In differential geometry, a subfield of mathematics, the Margulis lemma is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifolds. Roughly, it states that within a fixed radius, usually called the Margulis constant, the structure of the orbits of such a group cannot be too complicated. More precisely, within this radius around a point all points in its orbit are in fact in the orbit of a nilpotent subgroup.

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This is a glossary of algebraic geometry.

Quadrisecant

In geometry, a quadrisecant or quadrisecant line of a curve is a line that passes through four points of the curve. Every knotted curve in three-dimensional Euclidean space has a quadrisecant. The number of quadrisecants of an algebraic curve in complex projective space can be computed by a formula derived by Arthur Cayley. Quadrisecants of skew lines are also associated with ruled surfaces and the Schläfli double six configuration.

In real algebraic geometry, Gudkov's conjecture, also called Gudkov’s congruence, was a conjecture, and is now a theorem, which states that an M-curve of even degree obeys the congruence

References

Phillip Griffiths American mathematician

Phillip Augustus Griffiths IV is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particular of the theory of variation of Hodge structure in Hodge theory and moduli theory.

Joe Harris (mathematician) American mathematician

Joseph Daniel Harris, known nearly universally as Joe Harris, is a mathematician at Harvard University working in the field of algebraic geometry. He attended college at and received his Ph.D. from Harvard in 1978 under Phillip Griffiths.