Could anyone here give me some reference on finding defining equations for secant variety of any projective varieties? Ex: secant variety of Segre, Grassmanian ...

## 6 Answers

There is an advanced monograph by F.L.ZAK on exactly this subject: "Tangents and secants of algebraic Varieties" (AMS, 1993, Translations of Mathematical Monographs 127). A free pdf version is online

http://mathecon.cemi.rssi.ru/zak/files/Zak_TSAV.pdf

Charles's excellent reference is more elementary, so according to the level of technicity you are ready to tolerate you may navigate between both books.

Harris's book on Algebraic Geometry has a lot of this in it. Try lectures 8 and 9, in 8 he introduces secant varieties and makes the first two nontrivial examples exercises (do them. It will help) and in 9 he talks about computing the secant variety of a determinantal variety.

You could also try out Brendan Hassett's book "Introduction to Algebraic Geometry". In the fourth chapter he introduces secant varieties and even computes some examples using the techniques provided by Gröbner Bases. It is a really nice book.

The equations that cut out the first secant variety to a Segre variety were computed in characteristic 0 by Claudiu Raicu: http://arxiv.org/abs/1011.5867

I'm very late to the conversation. In general nothing is known. For some cases of the Segre or Veronese variety, one can interpet the varieties as spaces of matrices and then the equations are determinants. In general this is not known. There is a large current literature on this topic. I would start at the arxiv with pretty much any current paper by J.M. Landsberg. It will contain loads of information and references on this topic.