HOMEWORK 4
This problem set is due Monday September 29. 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. A polynomial is completely reducible if it is a product of linear terms. Prove that the
locus of completely reducible polynomials of degree din n+1 variables in P(n+d
d)−1is a projective variety.
Similarly, prove that polynomials that are d-th powers of linear forms (Ld)form a projective variety.
Prove that this variety is the d-th Veronese embedding of Pn.
Problem 0.2. Let the dual projective space Pn∗denote the space of hyperplanes in Pn. Show that the
universal hyperplane
Γ = {(H, p) : p∈H} ⊂ Pn∗×Pn
(i.e. the pairs consisting of a hyperplane and a point contained in that hyperplane) is a projective variety.
Prove, in fact, that it is a hyperplane section of the Segre variety Pn∗×Pn⊂Pn2+2n. What is its
dimension? Let Xbe a projective variety. Prove that the universal hyperplane section of Xdefined as
ΓX={(H, p) : p∈H∩X} ⊂ Pn∗×Pn
is a projective variety. Calculate the dimension of ΓXin terms of the dimension of Xand n.
Problem 0.3. Let P(n+d
d)−1denote the parameter space of hypersurfaces of degree din n+ 1 variables.
Show that the universal hypersurface
Ωd={(F, p) : F(p) = 0} ⊂ P(n+d
d)−1×Pn
is a projective variety. What is this variety when d= 1? Calculate the dimension of Ωd.
Problem 0.4. For this problem assume that k=C. We say that a hypersurface defined by a polynomial
F= 0 is singular at a point pif
F(p) = Fx0(p) = · · · =Fxn(p)=0
Fand all its first order partial derivatives vanish at p. Prove that a quadratic polynomial in n+1 variables
is singular if and only if the determinant of the associated symmetric matrix is zero.
Problem 0.5. Show that the locus of homogeneous polynomials of degree din n+ 1 variables that have
a singular point is a projective variety. Show that it has codimension one in the space of all polynomials
of degree din n+ 1 variables. Hence, it can be described as the zero locus of a single polynomial in n+d
d
variables. Show that if d= 2, then the degree of this polynomial is n+ 1. Chal lenge: What is the degree
of this polynomial for arbitrary d?
Problem 0.6. Show that a general hypersurface of degree d > 2n−3in Pndoes not contain any lines.
Generalize this statement from lines to linear spaces of higher dimension.
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