# how to compute the dimension of secant variety

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.
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 $\dim Hilb_2(X) + 1 = 2\dim X + 1$.