Douady rabbit: Difference between revisions

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In algebraic geometry, the Zariski closure of the union of the secant lines to a projective variety $X\subset \mathbb {P} ^{n}$ is the first secant variety to $X$ . It is usually denoted $\Sigma _{1}$ .

The $k^{th}$ secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on $X$ . It is usually denoted $\Sigma _{k}$ . Unless $\Sigma _{k}=\mathbb {P} ^{n}$ , it is always singular along $\Sigma _{k-1}$ , but may have other singular points.

If $X$ has dimension d, the dimension of $\Sigma _{k}$ is at most kd+d+k.

References

Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York. ISBN 0-387-97716-3

Template:Algebra-stub

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+In [[algebraic geometry]], the [[Zariski closure]] of the union of the [[secant line]]s to a [[projective variety]] <math> X\subset\mathbb{P}^n </math> is the first '''secant variety''' to <math> X </math>.  It is usually denoted <math>\Sigma_1</math>.
+The '''<math>k^{th}</math> secant variety''' is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on <math>X</math>.  It is usually denoted <math>\Sigma_k</math>.  Unless <math>\Sigma_k=\mathbb{P}^n</math>, it is always singular along <math>\Sigma_{k-1}</math>, but may have other singular points.
+If <math>X</math> has dimension d, the dimension of <math>\Sigma_k</math> is at most kd+d+k.
+==References==
+* Joe Harris, ''Algebraic Geometry, A First Course'', (1992) Springer-Verlag, New York. ISBN 0-387-97716-3
+[[Category:Algebraic geometry]]
+{{algebra-stub}}
+{{geometry-stub}}

Douady rabbit: Difference between revisions

Revision as of 00:21, 26 October 2013

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