(1) Let X n ⊂ PV = PN be a projective variety, and recall the definition of the r-th secant
variety σr (X) ⊂ PV from class. Show that dim σr (X) = rn+r −1 if there exists a point of
some Pr−1 spanned by r points of X that lies in no other Pr−1 spanned by r points of X.
In particular, show that if n = 1, then unless X is contained in a plane, dim σ2 (X) = 3.
(2) Assuming rn + r − 1 ≤ N , define the secant defect δσ,r (X) to be rn + r − 1 − dim σr (X).
Show that δσ,2 (Seg(P2 × P2 )) = 2.
(3) Let X ⊂ PV = PN be an irreducible projective variety. Define the dual variety of X,
X ∨ ⊂ PV ∗ , by
X ∨ = {H ∈ PV ∗ ∣ ∃x ∈ Xsmooth such that x ∈ (X ∩ H)sing }.
Construct an incidence correspondence in F1,N (V ) and use it to show i) X ∨ is irreducible,
ii) if X is a curve not contained in a hyperplane, then X ∨ is a hypersurface, and iii)
(X ∨ )∨ = X.
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