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Unformatted text preview: NOTES ON THE ARITHMETIC OF ALGEBRAIC CURVES PETE L. CLARK 1. What’s Nice About Algebraic Curves In this course we are going to study algebraic curves – i.e., one-dimensional alge- braic varieties – and how they vary in one-parameter families. In fact curves are by far the most intensively studied class of algebraic varieties (and this is true pretty much across the board, from complex algebraic geome- ters to arithmetic geometers). 1 Let us begin by recalling some nice properties that curves have and most higher-dimensional varieties lack. 1. Topological classification: Here let us work, for simplicity, over the complex numbers (although with suitable technology, one can establish quite similar results in positive characteristic). Any smooth, projective connected algebraic variety over C determines a complex manifold, whose complex dimension is equal to the alge- braic dimension of the variety and therefore whose dimension as a real manifold is equal to twice that. Therefore a complex algebraic curve gives rise to a compact topological surface, whereas a complex surface gives rise to a compact topological 4-manifold. But the jump in topological complexity between 2-dimensional mani- folds and 4-dimensional manifolds is extreme. Every (orientable, as all C-manifolds are) compact surface is homeomorphic to a g-holed torus (a 0-holed torus being a sphere). Thus there is a single topological invariant, the genus g . Moreover the topology of a genus g surface is well-understood: in particular its fundamental group is known to be Π( g ) = h a 1 ,b 1 ,...,a g ,b g | [ a 1 ,b 1 ] ··· [ a g ,b g ] = 1 i . On the other hand, every finitely presented group is the fundamental group of some compact topological 4-manifold. Being the fundamental group of an alge- braic complex manifold brings some additional restrictions: for instance, Hodge theory implies that its abelianization is of the form Z 2 n ⊕ T where T is a finite abelian group – i.e., its first Betti number must be even. In fact it can be shown that a finitely presented group is the fundamental group of an algebraic surface iff it is the fundamental group of any smooth algebraic variety (iff it is the fundamental group of a compact K¨ ahler manifold). Exactly what class of groups this might be is a fascinating open problem: indeed there is a book called Fundamental Groups of Compact K¨ ahler Manifolds (which does not come close to solving the prob- lem, but presents lots of interesting results). This observation has important geometric consequences: when one studies fam- ilies of algebraic varieties – roughly speaking, moduli spaces – in order to for the 1 Probably second place goes to abelian varieties, in part because of their close relationship with curves....
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This note was uploaded on 10/26/2011 for the course MATH 8320 taught by Professor Clark during the Fall '10 term at UGA.
- Fall '10