Secant variety explained

In algebraic geometry, the secant variety

\operatorname{Sect}(V) , or the variety of chords, of a projective variety V \subset P ^{r} is the Zariski closure of the union of all secant lines (chords) to V in P ^{r} :\operatorname{Sect}(V) = c up _{x,} \overline{xy} (for x = y , the line \overline{xy} is the tangent line.) It is also the image under the projection p _{3:} ( P ^{r)} ^{3} \to 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 ^{r} _{k = P} , 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

P ^{3} as follows. Let C \subset 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 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 P ^{r-1} . Now repeat.

If

S \subset P ^{5} is a surface that does not lie in a hyperplane and if \operatorname{Sect}(S) \ne P ^{5} , then S is a Veronese surface.

References

• • Book: Phillip Griffiths

. P. . Griffiths . Phillip Griffiths . J. . Harris . Joe Harris (mathematician) . Principles of Algebraic Geometry . Wiley Classics Library . Wiley Interscience . 1994 . 0-471-05059-8 . 617 .

• Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York.