2012 Math 631
Professor Karen Smith
Quiz 1
1. TRUE OR FALSE: The integers Z form a closed subset of the real numbers R in the
Zariski topology.
[False: the proper closed sets are all nite.]
2. Let V = V(f1 , . . . , ft ) and W = V(g1 , . . . , gs ) be two
MATH 631: ALGEBRAIC GEOMETRY: HOMEWORK 4 SOLUTIONS
Problem 1. (a) = (b) Suppose : X Y is a morphism, and x X . We may assume U is
a neighborhood of x with (U ) U0 Pm such that
|U : U Y U0 U0
Am
is given by
y (1 (y ), . . . , m (y )
where
1 , . . . , m OX
MATH 631: ALGEBRAIC GEOMETRY: HOMEWORK 3 SOLUTIONS
Problem 1. (a.) Write a polynomial as a sum of its homogenous components: f = fd + fd+1 +
+ fn , where each fi is degree i. Then t k acts by t f = td fd + td+1 ff +1 + + tn fn . So,
f is homogeneous if a
MATH 631: ALGEBRAIC GEOMETRY: HOMEWORK 2 SOLUTIONS
Problem 1. It suces to show that an ane algebraic set V k n is irreducible if and only if
its ideal I(V ) is prime. Suppose V = V1 V2 , where V1 and V2 are proper closed subsets. Since
Vi
V , we know I(V
MATH 631: ALGEBRAIC GEOMETRY: HOMEWORK 1 SOLUTIONS
Problem 1. (a.) The (t + 1) (t + 1) minors m1 (A), . . . , mk (A) of an n m matrix A are
polynomials in the entries of A, and mi (A) = 0 for all i = 1, . . . , k if and only if rank(A) t. In
other words,
MATH 631: ALGEBRAIC GEOMETRY: HOMEWORK 6 SOLUTIONS
Problem 1. (a.) If X Pn is a projective variety, we will write C (X ) for the cone over X in
An+1 . The homogeneous radical ideal I(X ) dening X in Pn is equal to the ideal I(C (X ) dening
C (X ) in An+1
2012 Math 631
Professor Karen Smith
Quiz 7
TRUE OR FALSE:
1. If X Y is the blowup of the smooth afne variety Y along an ideal I , then X is smooth.
2. If X A2 is the blowup of the origin, then the ber over the origin is a prime divisor of X .
3. The divis
2012 Math 631
Professor Karen Smith
Quiz 6
Let Y1 , . . . , Yr be closed subvarieties of a smooth ambient variety Y , and suppose that p Y1 Yr .
By denition, the varieties Y1 , . . . Yr intersect transversely at p if the codimension of the vector subspace
2012 Math 631
Professor Karen Smith
Quiz 5
Let V = V(y 4 + y 2 + 10x3 + 2x2 ) C2 .
[ I am changing the signs so that p actually lies on V ! The equation is slightly different from the
one in class.]
1. Describe the tangent space to V at (0, 0).
2. Compute
2012 Math 631
Professor Karen Smith
Quiz 4
Consider the Segre map P1 P1 P3 sending
([X0 : X1 ], [Y0 : Y1 ]) [X0 Y0 : X0 Y1 : X1 Y0 : X1 Y1 ]).
Denote the image . Let us consider the set U0 U0 = A1 A1 = A2 in P1 P1 to be the
nite part of the product. The c
2012 Math 631
Professor Karen Smith
Quiz 3
The twisted cubic C in A3 can be dened by the two equations y x2 , z xy . Their
homogenizations W Y X 2 , ZW XY dene a projective algebraic set V in P3 .
1. Describe V UW by explicitly giving a set of dening poly
2012 Math 631
Professor Karen Smith
Quiz 2
1. Prove or disprove: Any two distinct lines in P2 intersect in a unique point. [Recall
k
that a line in a Pn is a set of one-dimensional subspaces of k n+1 which lie in a xed twok
dimensional subspace.]
[Solutio
MATH 631: ALGEBRAIC GEOMETRY: HOMEWORK 5 SOLUTIONS
Problem 1. (a.) Let x0 , . . . , xn denote the homogeneous coordinates of Pn , y0 , . . . , ym the homogeneous coordinates of Pm , and zij for i = 0, . . . , n and j = 0, . . . , m the homogeneous coordin