Math 619: Riemann Surfaces and Algebraic Curves
WHEN: MWF 10 – 10:50 pm
WHERE: WAGAR 107 B / In the oval with good weather
WHAT: Riemann Surfaces and Algebraic Curves
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Goal of this class is to explore the interconnections of
the analytic theory of Riemann Surfaces and the algebro-geometric
theory of Algebraic Curves. We may take a detour into combinatorics,
representation theory of the symmetric group, and maybe even a little
bit of physics, depending on how things evolve.
HOMEWORK
[SEP. 20] 1.
Prove that a bi-homogeneous polynomial F in X_0, X_1, Y_0,Y_1 defines a
Riemann Surface C in P^1xP^1 if and only if there is no point P
in C where F, one of the
partials with respect to the X's
and one of the partials with respect to the Y's vanish simultaneously.
2. Prove that a (projective) plane conic is isomorphic to P^1.
3. Prove (carefully) that functions P^1 -> P^1 are precisely
rational functions of one affine local coordinate, or,
alternatively, homogeneous rational functions of degree zero
in the homogeneous coordinates.
[OCT. 23] 1.
Consider a line bundle O(k) on P^1. Write down the transition function
for the trivialization of O(k) over the standard covering of P^1
consisting of the two open sets missing 0 and
infinity. When you restrict a section to either trivialization
you get a function. Show that the transition between these local
functions has opposite exponent than the one on the charts.
(explicitly, a section which is constant on one chart will become the
polynomial z^k on the other). Think of a philosophical explanation for
this phenomenon.
2. Prove that the cotangent bundle of P^1 is isomorphic to
O(-2), and the tangent bundle of P^1 is isomorphic to O(2).
3. The tautological bundle on P^1 is the subbundle of P^1x C^2 such
that the fiber of each point of P^1 is exactly the line parameterized
by the point. Write down equations for this condition. Show that this
line bundle is isomorphic to O(-1).
BOOKS:
There are three books I am thinking of using (parts of) for this class:
1· R. Miranda. Algebraic Curves and Riemann Surfaces.
2· R. Cavalieri, E. Miles. Riemann Surfaces and Algebraic Curves: a first course in Hurwitz Theory.
3· E. Arbarello, M. Cornalba, P. Griffiths, J. Harris. Algebraic Curves.