Math 145, Problem Set 6. Due Friday, May 23.
You may assume that the ground eld is k = C.
1. (Hyperelliptic curves.) Let a1 , . . . , a5 be pairwise distinct constants. Find the singularities of
the projective hyperelliptic curve of genus 2:
y 2 z 3 = (x
Math 145, Problem Set 2. Due Friday, April 18.
1. Let X1 , X2 be afne algebraic sets in An . Show that
(i) I (X1 X2 ) = I (X1 ) I (X2 ),
(ii) I (X1 X2 ) = I (X1 ) + I (X2 ).
Show by example that taking the radical in (ii) is in general necessary, i.e. nd
Math 145, Problem Set 3. Due Friday, April 25.
For this problem set, you may assume that the ground eld is k = C.
1. Find the irreducible components of the afne algebraic set x2 yz = xz x = 0 in A3 .
2. Find the irreducible components of the afne algebrai
Math 145, Problem Set 4. Due Friday, May 2.
For this problem set, you may assume that the ground eld is k = C.
1. Let f and g be distinct irreducible polynomials in k [X, Y ] of degrees d and e. Show that
Z (f ) and Z (g ) intersect in at most de points:
Math 145, Problem Set 5. Due Friday, May 16.
You may assume that the ground eld is k = C.
1. (Products of afne varieties.) Let X An and Y Am be afne algebraic sets.
(i) Show that X Y An+m is also an afne algebraic set.
(ii) Extra credit and entirely optio