School of Mathematics
Module MA2318 - Elementary projective and algebraic geometry
2011-12 (SF & JS Mathematics
)
Lecturer: Prof. R. Tange
Requirements/prerequisites: MA1111/1212, MA2215, MA2321, MA2325
Duration: Hilary term, 11 weeks
Number of lectures per week: 3 lectures including tutorials per week
Assessment: Regular assignments.
ECTS credits: 5
End-of-year Examination:
2 hour examination in Trinity term.
Description:
Textbook: Miles Reid Undergraduate Algebraic Geometry,
London Mathematical Society Student Texts.
Tentative syllabus:
Algebraic curves. Conics (or quadrics), their euclidean and affine
classification over the fields of real and complex numbers. Projective
plane, homogeneous coordinates, projective transformations. Lines in
projective plane. Projective classification of conics. Parametrization
of nondegenerate conics.
Homogeneous polynomials or forms. Roots of polynomials and their
multiplicities. Bezout's Theorem, proof when one of the curve is a
line or a quadric. Factorization of forms vanishing along lines and
nondegenerate conics. Spaces of d-forms vanishing at certain points
and their dimensions. Applications to quadrics passing through 5
points and cubics passing through 9 points. Pascal's Theorem.
Nodal and cuspidal cubics, their parametrization. Tangent lines. Group
law on a cubic. Riemann surfaces and genus.
Affine algebraic sets, their ideals. Noetherian rings. Hilbert Basis
Theorem. Algebraic sets defined by ideals, their properties. Zariski
topology. Termination of descending chains of algebraic sets.
Irreducible algebraic sets, their relation with prime ideals. Unique
decomposition of algebraic set into irreducible components.
Nullstellensatz (Hilber Zero Theorem) and Weak Nullstellensatz. Proof
of the Nullstellensatz assuming Weak Nullstellensatz.
Polynomial functions on affine algebraic sets. Coordinate ring.
Polynomials maps between affine algebraic sets. Relation between
polynomial maps and coordinate ring homomorphisms. Polynomial
isomorphisms. Affine varieties. Rational functions on affine algebraic
sets. Regular points of rational functions. Rational maps. Dominant
maps.
Projective algebraic sets. Homogeneous ideals and correspondence
between them and projective algebraic sets. The affine cone over a
projective algebraic set. Rational functions on projective algebraic
sets, rational maps between them. Regular points of rational functions
and maps. Morphisms and isomorphisms. Segre embedding of the product
of two projective spaces into another projective space. Finite unions,
finite products, and arbitrary intersections of projective algebraic
sets are again projective algebraic.
Learning
Outcomes:
On successful completion of this module, students will be able to:
- Give the affine classification of quadrics over \mathbbR or
\mathbbC (you may assume the degree is 2, not < 2).
Give the projective classification of quadrics over \mathbbR or
\mathbbC.
- State Bezout's Theorem (the multiplicities do not have to be
defined) and prove it in the case one of the curves is a line or a
nondegenerate quadric using the standard parameterisation of a
nondegenerate quadric and the result on factorisation of homogeneous
polynomials in two variables. Use Bezout's Theorem to show that there
exists a unique quadric passing trough 5 distinct points in
\mathbbP2 no 4 of which lie on a line.
- Determine the projectivisation, the point(s) at infinity, the
singular (= not smooth) points of a plane curve. Check that a point of
a curve is an inflection point.
- Explain what is meant by an affine or a projective algebraic set,
when such a set is irreducible and what is meant by the irreducible
components of such a set. Define what a (homogeneous) ideal is and
describe the variety-ideal correspondence. State Hilbert's Basis
Theorem and Hilbert's (Strong) Nullstellensatz.
- Give the construction of the field of rational functions on an
irreducible affine or projective algebraic set. Explain what the
domain of a rational function is. Explain what a rational map or a
morphism is between two affine or projective algebraic sets. Explain
what is meant by the Segre embedding and indicate how it can be used
to show that the product of two projective algebraic sets is again a
projective algebraic set.
Apr 1, 2012
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On 1 Apr 2012, 16:43.