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Unformatted text preview: FIRST STEPS IN THE GEOMETRY OF CURVES PETE L. CLARK We would now like to begin studying algebraic curves. However, to do this from the desired modern perspective we still need some further tools. Perhaps though by trying to study curves anyway the need for these specific tools will become clear. Let C /k be a geometrically integral curve. We have already seen that there ex- ists a unique curve C /k which is birational to C and is projective and nonsingular (and that this cannot be improved to smooth if k is not perfect). Thus the bira- tional geometry of curves works out in the nicest possible way. One often takes advantage of this by defining a curve using a model which is incomplete, singular or both (but always geometrically integral!), with the understanding that what we are really interested in is the unique projective nonsingular model. Perhaps the first thing we wish to define on algebraic curve is its genus , g N . Classically the genus was viewed as a topological invariant. Namely, if C / C is a nice curve, then C ( C ) in the analytic topology is a compact complex manifold of dimen- sion 1, so in particular a compact orientable real surface, and thus diffeomorphic to a sphere with g handles for some unique non-negative integer g . (Alternately, 2 g = dim Q H 1 ( C ( C ) , Q ).) For a nice curve over an arbitrary field k we could, in fact, make this topolog- ical definition work, but only with more sophistication and work than would be necessary to give a purely algebraic definition. Perhaps the cleanest algebraic definition of the genus is dim k H 1 ( C, O C ), i.e., the dimension of the first sheaf cohomology group of the structure sheaf. This points out the need for sheaf cohomology, which we have not yet discussed. Another useful algebraic definition dual to the first is dim k H ( C, C ). This is easier than the previous definition in that we are not really using cohomology: it is just the dimension of the space of global sections of a certain sheaf C , the canonical sheaf . However, to define this sheaf requires the notion of differentials. The next issue is how to embed a curve into projective space. For this we need the theory of divisors, line bundles and linear systems, and the Riemann-Roch theorem. Let us come at this from a very naive perspective. Let X be a variety which is geometrically integral and nonsingular in codimen- sion 1 (e.g., normal). For 0 i d = dim( X ), we will define an abelian group Z i ( X ). By a prime cycle , we mean a reduced irreducible closed subvariety of X . Like every irreducible variety, a prime cycle has a well-defined dimension, and we 1 2 PETE L. CLARK put Z i ( X ) to be the free abelian group generated by the prime cycles of dimension i . An element of the group Z d- 1 ( X ), of cycles of codimension one, is called a Weil divisor ....
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