HOMEWORK 5
This problem set is due Monday October 6. You may work on the problem set in groups; however,
the final write-up must be yours and reflect your own understanding. In all these exercises assume that
kis an algebraically closed field and Ris a commutative ring with unit.
Problem 0.1. Recall that “If f:X→Yis a surjective morphism of projective varieties such that
(1) Yis irreducible,
(2) Every fiber of fis irreducible,
(3) Every fiber of fhas the same dimension,
then Xis irreducible.” Show that all three assumptions are necessary.
Problem 0.2. Compute the multiplication table for the cohomology of G(2,5).
Problem 0.3. Prove Pieri’s formula
σ1·σλ1,...,λk=X
λi≤µi≤λi−1,Pµi=1+Pλi
σµ1,...,µk
where σλ1,...,λkand σµ1,...,µkare Schubert cycles in G(k, n).
Problem 0.4. We say that a plane curve F= 0 has a cusp at pif the Taylor expansion of Fat phas
the form
L2+h.o.t.
where Lis a line containing pand h.o.t. denotes higher order terms. Show that for d > 2plane curves
of degree dthat have a cusp form a projective subvariety of codimension two in Pd(d+3)/2, the space of
plane curves of degree d. (Hint: Linearize the problem by considering plane curves that have a cusp at p
with tangent direction L.)
Problem 0.5. Let X⊂Pnbe a projective variety. The secant variety to Xis the closure of the union
of lines spanned by distinct points on X
Sec(X) = ∪p,q∈X,p6=qpq .
Prove that Sec(X)is a projective variety of dimension less than or equal to min(2 dim(X) + 1, n).We
say that the secant variety is defective if dim(Sec(X)) <min(2 dim(X)+1, n). Prove that Sec(X)is
defective if and only if every point x∈Sec(X)lies on infinitely many secant lines to X. Show that the
secant variety of the Veronese image ν2(P2)in P5is defective. Hard Challenge: Show that a surface Sin
P5which is not contained in any hyperplane has a defective secant variety if and only if Sis the Veronese
image ν2(P2).
Problem 0.6. More generally, let X⊂Pnbe a projective variety. The r-secant variety Secr(X)to Xis
the closure of the union of the Pr−1’s spanned by rdistinct points p1,. . . , prin Xin general linear position.
Prove that Secr(X)is a projective variety of dimension less than or equal to min(rdim(X)+ r−1, n). We
say that Secr(X)is defective if the dimension of Secr(X)is strictly less than min(rdim(X) + r−1, n).
Show that Secr(X)is defective if and only if every point on Secr(X)is contained in infinitely many
secant Pr−1’s to X. Show that the fourth Veronese image ν4(P2)⊂P14 has a defective 5-secant variety
Sec5(ν4(P2)). Hard Challenge: Show that among the secant varieties to the Veronese images of P2,
Sec2(ν2(P2)) and S ec5(ν4(P2)) are the only defective secant varieties.
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