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By L. D. Olson

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Example text

We obtain a family of stable curves. (V Away from s = 0 to the point 0 - Spec

2 6. 's We already know their g-l,g-l . Therefore, we may intersect both D .. = aX + 2 b. 6, g-l,g-l I I section 4) and D F. (which we know exists from ^ and obtain linear relations among the co- We get enough independent relations to evaluate It is simply a matter of solving [y] + 2 linear unknowns. First let us state the three lemmas we will prove, leaving their proofs until later. 2) Lemma: D . 3) Lemma: ^g_lig_1 . • F3 = f-g(g-l)2 (g-2) (2g-5) + g(g-l)(g-2) 2 Applying these lemmas as explained in the first paragraph of this section one easily proves.

2) as g- g(g-l) (g-2) (2g-5) . 8) q is . 44 STEVEN DIAZ Case 2. C The g - 1 must occupy all fold point is over g - 1 tion point. Then s (g-2)(g-l)g choices for sheets. p s Over D and one other point of choices for away from filled with once. q r . C . be the Since p p x g - 1 C will be , q occurs, so over 2 . r . there are For each choice of sheeted over g - 3 C fold ramificaThere are was generically chosen there is no D C. s g - 1 which meets there are D are rational curves, all are unramified and each meets (g-2) = g(g-l)(g-2) 2 .

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