ALGEBRAIC CURVES (MA30188): 2003 EXAM
Retyped hastily after the original was lost. No guarantees, and no solutions either.
1. Describe the group law on a smooth plane cubic E⊂P2with a given point O∈Eas
origin.
Let E⊂P2be the curve over Cgiven by
y2z=x(x+ 3z)(x+ 27z)
and take Oto be the point (−9: −18√3: 1). What is the tangent line to Eat O? Show
that it meets Eagain at (0 : 0 : 1).
There are exactly four points A∈Esuch that A+A=O. Find them.
2. What does it mean to say that a commutativ ring is Noetherian?
State and prove Hilbert’s Basis Theorem.
If Z100 denotes the ring of integers modulo 100, is Z100[X] a Noetherian ring? Give reasons
for your answer.
3. Define the tangent space TPVto a hypersurface V⊆Anin affine space at a point
P∈V. What does it mean to say that Pis a singular point of V?
Show that if the ground field kis algebraically closed then the set of non-singular points
of Vis non-empty. [Hilbert’s Nullstellensatz may be assumed.]
Find the singular points of the astroid, which is the curve in A2over k=Cgiven by
(x2+y2+ 1)3+ 27x2y2= 0.
4. What is meant by a rational map between projective algebraic varieties? What is
meant by a morphism of algebraic varieties? What does it man to say that two varieties
are birationally equivalent? What does it mean to say that they are isomorphic?
Show that the rational map P299K P2given by
(x:y:z)→(yz :zx :xy )
has a rational inverse map.
Show that the projective curves with equations
y2z−x(x−z)(x−2z)=0
and
x3z−y2z2+ 3xy2z−2x2y2
are birationally equivalent.