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This is the Past Exam Paper of Math Tripos which includes Boundary Value Problems for Integrable Pde's, Black Holes, Biostatistics, Basic Algebraic Geometry etc. Key important points are: Basic Algebraic Geometry, Nonsingular Cubic Surface, Dimension Theory of Morphisms, Algebraic Varieties, Rational Normal Curve, Homogeneous Ideal, Three Obvious Quadrics, Field of Fractions, Cech Cohomology
Typology: Exams
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Thursday 29 May 2003 9 to 12
Attempt THREE questions.
There are five questions in total. The questions carry equal weight.
1 Prove carefully that a nonsingular cubic surface contains a line. You may state without proof any results from the dimension theory of morphisms of algebraic varieties.
2 Let X ⊂ P^3 be the rational normal curve in P^3 ; in other words, X is the image of the morphism P^1 3 (s, t) → (s^3 , s^2 t, st^2 , t^3 ) ∈ P^3
Prove that the homogeneous ideal of X is the ideal
I = (x 0 x 3 − x 1 x 2 , x 0 x 2 − x^21 , x 1 x 3 − x^22 )
generated by the three obvious quadrics containing X.
3 Let A be a unique factorization domain, K its field of fractions. Let K ⊂ L be an algebraic extension of fields. Then if b ∈ L is integral over A, the norm N (^) KL (b) is an element of A. Discuss briefly the relevance of this statement to the dimension theory of algebraic varieties.
4 Write an essay on coherent cohomology. Your essay should at least contain the definition of Cech cohomology, state some basic theorems on the cohomology of coherentˇ sheaves on projective space and projective varieties, and discuss some examples.
5 State briefly the axiomatic properties of Chern classes of vector bundles on (nonsingular) algebraic varieties.
If Y is a nonsingular algebraic variety, denote as usual by Ω^1 Y the cotangent bundle of Y (this is the same as the bundle of Kaehler differentials). Let Xd ⊂ Pn^ be a nonsingular hypersurface of degree d in Pn; calculate the Chern classes ci(Ω^1 X ) in terms of h = c 1 (OX (1)).
[Use the exact sequences 0 → Ω^1 P → OP(−1)n+1^ → OP → 0 , and 0 → OX (−d) → Ω^1 P|X → Ω^1 X → 0. Apply the Whitney formula.]
Paper 15
