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**Sample text**

Daniel 073 = 555 We can use #? to test whether a given key occurs in the hash table. c 075 = false Finite sets are implemented in Macaulay 2 as hash tables: the elements of the set are stored as the keys in the hash table, with the accompanying values all being 1. a 077 = true i78 : peek x 078 = Set{{4, 5} => 1} 1 => 1 a => 1 i79 : y = tally{1,a,{4,5},a} 079 = Tally{{4, 5} => 1} 1 => 1 a => 2 079 Tally i80 y#a 080 = 2 We might use tally to tally how often a function attains its various possible values.

32 D. Eisenbud For a curve of genus 6 such as X, the gonality is either 2 (the hyperelliptic case), 3 (the trigonal case) or 4 (the value for general curves). The Clifford index, on the other hand is either 0 (the hyperelliptic case) or 1 (the case of a trigonal curve OR a smooth plane quintic curve-which is necessarily of gonality 4) or 2 (the case of a general curve). Thus for most curves (and this is true in any genus) the Clifford index is equal to the gonality minus 2. ) To make further progress we use an idea of Mark Green (see Green and Lazarsfeld [5]).

We next ask for analytic information about the curve and the embedding. A reasonable place to start is with the relation between the line bundle defining the embedding and the canonical sheaf wx. Notice first that the degree of the hyperplane divisor (the degree of the curve) is 10 = 2g-2, the same as the canonical bundle. By Riemann-Roch the embedding line bundle either is the canonical bundle or has first cohomology 0, which we can check with i64 : P3 = Proj ringP3 064 = P3 064 Projective Variety i65 HH~1«OO_P3(1)/idealX)(>=0)) 1 065 : ringP3-module, quotient of ringP3 Let's examine the degree of the generator of that module.