Download Algebraic Geometry: Irreducibility, Cohomology, Schubert Cycles, Curves, Secant Varieties and more Assignments Mathematics in PDF only on Docsity! 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 k is an algebraically closed field and R is a commutative ring with unit. Problem 0.1. Recall that “If f : X → Y is a surjective morphism of projective varieties such that (1) Y is irreducible, (2) Every fiber of f is irreducible, (3) Every fiber of f has the same dimension, then X is 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 = ∑ λi≤µi≤λi−1, P µi=1+ P λi σµ1,...,µk where σλ1,...,λk and σµ1,...,µk are Schubert cycles in G(k, n). Problem 0.4. We say that a plane curve F = 0 has a cusp at p if the Taylor expansion of F at p has the form L2 + h.o.t. where L is a line containing p and h.o.t. denotes higher order terms. Show that for d > 2 plane curves of degree d that 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 ⊂ Pn be a projective variety. The secant variety to X is the closure of the union of lines spanned by distinct points on X Sec(X) = ∪p,q∈X,p 6=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 P5 is defective. Hard Challenge: Show that a surface S in P5 which is not contained in any hyperplane has a defective secant variety if and only if S is the Veronese image ν2(P2). Problem 0.6. More generally, let X ⊂ Pn be a projective variety. The r-secant variety Secr(X) to X is the closure of the union of the Pr−1’s spanned by r distinct points p1, . . . , pr in X in general linear position. Prove that Secr(X) is a projective variety of dimension less than or equal to min(r dim(X)+r−1, n). We say that Secr(X) is defective if the dimension of Secr(X) is strictly less than min(r dim(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 Sec5(ν4(P2)) are the only defective secant varieties. 1
