MATH 552: PROBLEM SET 4
Due Wednesday,October 12
(1) [Ha] Ch.I: 2.9, 2.11, 2.12, 2.14, 3.14
(2) Let X⊂Pmbe a quasi-projective variety and φ:X→Pna regular map.
Show that for all x0∈X, there exists an open neighborhood Uof x0and
homogeneous polynomials F0, . . . , Fnof the same degree which don’t vanish
simultaneously at every point of U, such that
φ(x)=(F0(x) : . . . :Fn(x)) on U.
Give an example of regular function which cannot be written like this on
the whole X.
(3) A conic in P2is a projective variety that can be written as the zero locus
of a single irreducible homogeneous polynomial of degree 2. Let Vbe the
vector space over kof homogeneous degree 2 polynomials in three variables,
and let P(V)∼
=P5be its projectivization.
(a) Show that the space of conics in P2can be identified with an open
subset U⊂P5. (We say that Uis a moduli space for conics in P2
and that P5is its compactification.) What geometric objects can be
associated with the points in P5−U?
(b) Show that the condition for a conic to pass through a given point is
a linear one; namely, if P∈P2, show that there is a linear subspace
L⊂P5such that the conics passing through Pare exactly those in
U∩L. What do the points in (P5−U)∩Lcorrespond to?
(c) Show that there is a unique conic passing through any five points in
P2, as long as no three of them lie on a line. What happens if three of
them do lie on a line?
(4) Extra credit problems. The following set of problems deals with the
important notion of a normal variety. I encourage you to look at this and
turn in what you can. There is no specific deadline.
[Ha] Ch.I: 3.17, 3.18, 3.20
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