By I.G. Macdonald

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**Extra resources for Algebraic geometry : introduction to schemes**

**Sample text**

Cl 14e eventually see that the picture of the limit of ~d(g,

Which are therefore divided into four groups, according to the four possibilities for the values of the vanishing sequences at R. Similarly each of the Pi's of the latter group is attached in one point to the copy of sdl(g-2,c~+2). These are therefore exceptional curves and can be contracted. Each of the Pi's of the second group is attached in one point to S~_2(g-2,c~-2) and to both copies of sl_l(g-2,~x) appearing in the limit. Similarly each of the Pl's of the third group is attached in $1 + I one point to d(g-2,

354. Then t h e r e is l- a scheme ~d(X)-~B over B whose points parametrize limit linear series of degree d and dimension r on the curve of ~r: X ~ B . Moreover ~ ( X ) - ~ B is flat over B if B is a smooth curve, the general fibre of it: X - ~ B is smooth and the special fibre is a curve of the type s h o w n in Fig. i. Proof. 3 in [EH1], w h e r e the existence of the scheme ~ ( X ) - ~ B is proved. Let B be a smooth curve and let q be a closed point of B such that the fibre Xq of it: X - , B is of compact type with n nodes.

### Algebraic geometry : introduction to schemes by I.G. Macdonald

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