By D. Burns (auth.), I. Dolgachev (eds.)

ISBN-10: 3540123377

ISBN-13: 9783540123378

ISBN-10: 3540409718

ISBN-13: 9783540409717

**Read or Download Algebraic Geometry: Proceedings of the Third Midwest Algebraic Geometry Conference held at the University of Michigan, Ann Arbor, USA, November 14–15, 1981 PDF**

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**Extra 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**

**Example text**

2. ,~a(Pi~) lie in a hyperplane Hi H.. 1 a e Ar is 6ood. Then a is an isolated component of i__nn A r. y-l(~(a)) Proof. The set of y(a) = ~ b ) and ~ Thus we may choose b Sa common component. in Ar so that is very ample. ~b so that Sa(2p(a)) Sa(Pij) = ~b(P~j), 2p(a) and is very ample is open. Suppose Then HO(2p(a),~a ) = HO(2p(b),~b) ~ HO(c,L). 2~b ) and for ~b(~p(b)) have j = i ..... r, regarded as elements of C where C. as a Pij The and Sa(Pij) are in general position in a hyperplane curve of degree dO r = d O + i - go' Thus given a, and genus we see the H..

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 [1]). h, and locally, ~(S) k nodes. is contained in any $(S). T. eodim~ W <_ p. Let W be a component of Br which First let us define a functor D F For eacN an effective relative Cartier divisor on S XM C so that @(D) is locally isomorphici > T' to the pullback of Using Grothendieck's Quot scheme, there is an M divisor F.

3 We get Y0 = w2' YI = -w0w~' expression of 4 in appropriate The system of cubics has and {w 2 = w I = 0} = P', 2 in Y0' Y2 = WoW]W2' coordinates on "'" Y3 and express then as homoge- 2 Y3 = w 0 ( w 0 w 2 - w l ) ' ~2 base points, namely and and this is an ~3 . {w 2 = w 0 =0} = P, and a general cubic of the system is smooth at P, P': 39 but to obtain a system free of base points one has to blow up three times over at the points where the line by {w 0 = 0} L0) passes and three times over 2 {w0w 2 - w I = 0} passes through.

### Algebraic Geometry: Proceedings of the Third Midwest Algebraic Geometry Conference held at the University of Michigan, Ann Arbor, USA, November 14–15, 1981 by D. Burns (auth.), I. Dolgachev (eds.)

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