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Additional resources for Algebraic Geometry: Proceedings of the Third Midwest Algebraic Geometry Conference held at the University of Michigan, Ann Arbor, USA, November 14–15, 1981
T We denote Now given with components to form a node. pl (t) ~ ~t C and There is a family and the map C X It] -~gt is the natural one. Now fix an integer ~t dO . We assume be the set of all line bundles and which have degree dO There is a variety on denotes the line bundle induced on construction of ~ and at ~ d O + 1 - go = r on ~t and d O > 2g 0 - 2. ,Csk. p : ~ ~T and every line bundle on ~ and a line bundle ~p(a) for occurs uniquely as to the reader. ~ a e ~, ha on ~ ×T ~ then each for some so that if ha a.
Suppose that the Brill-Noether condition at s V : X ~S and that is equinodal~ that £s is very ample. smooth non-de6enerate curves of desree d and 5enus g Proof. s. (w@3 ® £) of degree is free on d + 3(2g - 2) ~ Let Let g M in p : C ~6 ¢*(~). ~ : S -+M so that is just ¢-l(Mr) = S r. £. Further, if Suppose Ch M is stable. Ti T T of 7r : X -+ S T Mr ~(S). (w®3 ® £), L ® w ®3 is isomorphic to *. h Let be ~ ® w ®-3. then is singular] . is equinodal~ we see at h (Cfo ). h, and locally, ~(S) k nodes.
Are chosen generical~y~ 1 (i) (ii) then R i n Rj = #(R i N C) = r + 2. D = C U R I U ''" U P'k Further~ if and 2k > h0(C,L) - (r + i), we have hO(D,%(1)) = r + 1. Proof. Let for any fixed U ~, be the set of all hyperplanes meeting 7rl(U,~ ) C transversally. operates as the full symmetric group on ~ Then n C. It follows that if we set ~ then ~i = [(Pl ..... p~,H) is irreducible I H e U, PieHNC] , . Now F = [(Pl ..... c, b y hypothesis. is quasi-finite, we can choose the H. in HI , Since the map from not in the image of ~.