Could anyone here give me some reference on the dimension of secant variety? What is expected dimension? How to find out it is defective or not? Thank you very much.
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Dear mingming, you can find a lot of information on secant varieties in Harris's book "Algebraic Geometry, A First Course "(Springer GTM 133), essentially presented as a set of thoughtfully conceived exercises. The ultimate reference on the subject is ZAK's monograph
For those who don't know the concept yet let me briefly outline its basic idea. Given a $d$-dimensional variety $X$ in $\mathbb P^n$, take all the chords joining two points ( maybe not distinct: add tangents) of $X$ and consider the union $Sec(X)$ of these chords.This variety has dimension at most $2d+1$ and generically you have equality.This allows for many nice very geometric constructions, for example by projecting from a point outside $Sec(X)$ ( or from a linear subspace disjoint from $X$) . You can easily show this way that every projective smooth variety of dimension $d$ embeds in $ \mathbb P^{2d+1}$ or that every projective variety is birational to a hypersurface in projective space.
Edit Let me emphasize that the basic technique is very easy. For example, a secant line is determined by two points of $X$, so in the grassmannian of lines of $\mathbb P^n$ you get a $2d$-dimensional variety parametrizing the chords. Since the lines have dimension 1, you get the dimension $2d+1$ mentioned above for the secant variety. This looks very sloppy but the amazing and pleasant surprise is that a rigorous proof is pretty close to this sketch: cf. Harris's mentioned book, Proposition 11.24, page 144.
answered Apr 22, 2010 at 10:32
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$\begingroup$ Free pdf version of Zak's monograph: mathecon.cemi.rssi.ru/zak/files/Zak_TSAV.pdf $\endgroup$ Commented Apr 22, 2010 at 12:50
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Let $X \subset P(V)$. This map induces a map $Hilb_2(X) \to Hilb_2(P(V))$. On the other hand, there is a map $Hilb_2(P(V)) \to Gr(2,V)$ (a length 2 subscheme goes to the unique line conatining it). Let $U$ be the pullback to $Hilb_2(X)$ of the tautological rank 2 bundle on $Gr(2,V)$. Its projectivization $P_{Hilb_2(X)}(U)$ has a canonical map, its image is the secant variety. So, the expected dimension is $\dim Hilb_2(X) + 1 = 2\dim X + 1$.
