Secant variety#Examples

In algebraic geometry, the secant variety $\operatorname{Sect}(V)$ , or the variety of chords, of a projective variety $V \subset \mathbb{P}^r$ is the Zariski closure of the union of all secant lines (chords) to V in $\mathbb{P}^r$ :{{harvnb|Griffiths|Harris|1994|loc=pg. 173}}

: $\operatorname{Sect}(V) = \bigcup_{x, y \in V} \overline{xy}$

(for $x = y$ , the line $\overline{xy}$ is the tangent line.) It is also the image under the projection $p_3: (\mathbb{P}^r)^3 \to \mathbb{P}^r$ of the closure Z of the incidence variety

: $\{ (x, y, r) | x \wedge y \wedge r = 0 \}$ .

Note that Z has dimension $2 \dim V + 1$ and so $\operatorname{Sect}(V)$ has dimension at most $2 \dim V + 1$ .

More generally, the $k^{th}$ secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on $V$ . It may be denoted by $\Sigma_k$ . The above secant variety is the first secant variety. Unless $\Sigma_k=\mathbb{P}^r$ , it is always singular along $\Sigma_{k-1}$ , but may have other singular points.

If $V$ has dimension d, the dimension of $\Sigma_k$ is at most $kd+d+k$ .

A useful tool for computing the dimension of a secant variety is Terracini's lemma.

Examples

A secant variety can be used to show the fact that a smooth projective curve can be embedded into the projective 3-space $\mathbb{P}^3$ as follows.{{harvnb|Griffiths|Harris|1994|loc=pg. 215}} Let $C \subset \mathbb{P}^r$ be a smooth curve. Since the dimension of the secant variety S to C has dimension at most 3, if $r > 3$ , then there is a point p on $\mathbb{P}^r$ that is not on S and so we have the projection $\pi_p$ from p to a hyperplane H, which gives the embedding $\pi_p: C \hookrightarrow H \simeq \mathbb{P}^{r-1}$ . Now repeat.

If $S \subset \mathbb{P}^5$ is a surface that does not lie in a hyperplane and if $\operatorname{Sect}(S) \ne \mathbb{P}^5$ , then S is a Veronese surface.{{harvnb|Griffiths|Harris|1994|loc=pg. 179}}

Notes

References

{{citation|first=David|last=Eisenbud|first2=Harris|last2=Joe|title=3264 and All That: A Second Course in Algebraic Geometry|publisher=C. U.P.|year=2016|isbn=978-1107602724}}
{{cite book | first1=P. |last1=Griffiths | authorlink=Phillip Griffiths |first2=J. |last2=Harris |authorlink2=Joe Harris (mathematician) | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | page=617 }}
Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York. {{isbn|0-387-97716-3}}

Category:Algebraic geometry