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Algebraic Curves Exam - University of Bath, Exams of Algebra

Alliance UniversityAlgebra

The exam paper for the ma40188: algebraic curves module at the university of bath. It includes questions on commutative rings, affine varieties, polynomial maps, function fields, rational maps, and plane cubic curves. The exam is 2 hours long and calculators are not allowed. Only the best three answers will be assessed.

Typology: Exams

2012/2013

Uploaded on 02/18/2013

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MA40188
University of Bath
DEPARTMENT OF MATHEMATICAL SCIENCES
EXAMINATION
MA40188: ALGEBRAIC CURVES
Thursday 17th May 2007, 16.30–18.30
No calculators may be brought in and used.
Full marks will be given for correct answers to THREE questions.
Only the best three answers will contribute towards the assessment.
MA40188
pf3

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MA

University of Bath

DEPARTMENT OF MATHEMATICAL SCIENCES

EXAMINATION

MA40188: ALGEBRAIC CURVES

Thursday 17th May 2007, 16.30–18.

No calculators may be brought in and used.

Full marks will be given for correct answers to THREE questions. Only the best three answers will contribute towards the assessment.

MA

MA40188 2.

  1. (a) Define what is meant by an ideal of a commutative ring R. Define what it means for an ideal I of R to be a prime ideal. (b) Define what is meant by an affine variety over a field K. Say what it means for an affine variety V over K to be irreducible. (c) Say what is meant by a polynomial map f : V → W , where V and W are affine varieties. Explain how such a map induces a map f ∗^ : K[W ] → K[V ]. Show that if f ∗^ is injective and V is irreducible, then W is irreducible. (d) Now suppose K = C and let W ⊂ A^3 be the variety given by the three equations xz = y^2 , x = yz and y = z^2. By considering the map W → A^1 given by the z-coordinate, or otherwise, show that W is isomorphic to A^1. Deduce that W is irreducible. (e) Is the variety W ′^ ⊂ A^3 given by the two equations xz = y^2 and x = yz irreducible? Justify your answer briefly.
  2. (a) Define the function field K(V ) of an irreducible projective variety V over a field K. (b) Say what is meant by a rational map f : V 99K W , where W is a projective variety. What does it mean to say that f is regular at a point P ∈ V? What does it mean to say that f is dominating? What does it mean to say that f is birational? (c) What does it mean to say that an irreducible projective curve C over C is rational? What does it mean to say that an irreducible affine curve C 0 is rational? (d) Let C 0 be the curve in A^3 over C given by the equations xz = y and x = z^2 (z−1). By considering the intersection of C 0 with planes containing the z-axis, show that C 0 is rational. (e) Prove, using (d) or otherwise, that the curve C 1 ⊂ A^2 given by the equation x^4 = y^2 (y − x) is rational.

MA40188 continued