HOMEWORK 9
This problem set is due Monday November 3. You may work on the problem set in groups; however,
the final write-up must be yours and reflect your own understanding.
Problem 0.1. Calculate the degree of the Grassmannian of lines in Pnin its Pl¨ucker embedding. (Hint:
You may want to use Pieri’s formula and learn about Catalan numbers.)
Problem 0.2. Show that a variety Xand a general hyperplane section X∩Hhave the same degree.
Calculate the degree of the surface scroll in P4defined by the rank one locus of the matrix
z0z1z2
z2z3z4
Problem 0.3. Let a≤bbe two positive integers. The surface scroll Sa,b is constructed as follows. Pick
two disjoint linear spaces Paand Pbin Pa+b+1. Fix rational normal curves Caand Cbof degrees aand
bin the Paand Pb, respectively, and an isomorphism φ:Ca→Cb.Sa,b is the surface obtained by taking
the union of the lines joining p∈Cawith φ(p)in Cb. Prove that Sa,b is an algebraic surface. Describe in
detail S1,1. Show that the surface in the previous exercise is S1,2. We can also allow a= 0 and φto be
the constant map. In that case the surface is a cone over a rational normal curve. Calculate the degree
of the surface scroll Sa,b .
Problem 0.4. Let Xbe a non-degenerate (i.e., not contained in any hyperplanes) projective variety of
degree dand dimension kin Pn. Show that d≥n−k+ 1. Show that for rational normal curves, the
Veronese surface v2(P2)in P5and surface scrolls Sa,b equality holds. Challenge: Classify the varieties
where equality holds.
Problem 0.5. Prove that the every automorphism of projective space Pnis induced by an automorphism
φ∈GL(n+ 1) of kn+1. In other words, the automorphism group of Pnis PGL(n+ 1).
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