we have that - PDF document

1 ~ � 9 a � P 7 © �
~ � 9 a � P 7 © � Æ a § 4 a a 2 2 a a a & V 2 a § 4 the = · I 2 1 a a 7� 9 a 2 I a a — — - 2) - d - = 4 = the V^- a � a a - 2 (not two the P its V

2 @ ö 0 = is a V. ^ @ 2 + (X · 2H + K +
@ ö 0 = is a V. ^ @ 2 + (X · 2H + K + K 0 Ö + K ^ + K · D = (X · + 0 + K Ö a we have that V an a an a the · · ç / = , - n a 2, 2 a C 4) a 3) a W 1 0 C · is + l,d=2, a a a C a� Then

3 the P a If g the an the a a a one
the P a If g the an the a a a one the g P out on the V = a � the is is an is an have a Ð F a a C P a 2 a a a Let W de- the all Then W a a 2 a = W a a -I'm P a 4 a a = 2 2 2 a a = £

4 W is a = W Ã) P 2 a = ç a cut d d d
W is a = W Ã) P 2 a = ç a cut d d d 0 = 0 0 = , : · (V = is 5 · · � the the V: t ) a S S smooth cubic surface / = 4 ö Ö Ñ t = + bL\ q + b+ — 0 - Ê � - 5 6 1 0 3 0 - = \3M L\ 6L\ = \3M

5 - a a ® = 0 a W ® Ö | V' a a
- a a ® = 0 a W ® Ö | V' a a 2 d 1 + - l,a = 3 = {2 - Ó\ d = 2 - V a | I ^ = V of and ^ Æ è a 2 a� à = + S(V) V Æ Æ a a à ã = S Ö 0. Let C be a y à the are the m the the the the y, a

6 nd, in a y, � Ã� h Æ C
nd, in a y, � Ã� h Æ C a Æ Z\ Æ Ñ a / = a V' F Æ V' - ó' a - o = a Ð H' ó a of V a · -2. a Ö 0. Let ð be an S° be the let R° 2 is a V, and à 1 can a V, and the the a V. 2 2 e a 2

7 S° 2 a = = V; = ñ a = a a = Ö a
S° 2 a = = V; = ñ a = a a = Ö a C a a Å Æ. a a a a a a a a Ö Ë . ^ & S° a Æ C a the = Æ · is the à g a V g d is the g� not R + V 2 Æ · a Å V. a - = Æ · H, 4 a 4 a a F � r V

8 Æ a a F a + + 2. Æ 2 2 g� 4
Æ a a F a + + 2. Æ 2 2 g� 4 a F 4 a a Æ Ã) Ç = rZ + Ó^ C ( + ^ + 2 r — 2 + ÉÇ ä = 0 = m + 2 V' = - Z'\ - - = 1 4 a a a V' 2 a Æ, - r = V' — Z'). ~Z'. = D a = 2 cut p(S) � �

9 000; 2 a à the in + ... , the let Z
000; 2 a à the in + ... , the let Z' = a ~\Z) be i , + all i the on the � and no in a g 4 and no S Ô a t = ^ o\K + cut 4 a 2 = = V a a a a 6 6 2 a a a Ä = a = T(V) = q(T). A a on V Then

10 the normal ö ® a 0 a 0 · C) = = P
the normal ö ® a 0 a 0 · C) = = P(C) a • 4 O a C\ · Ç = V = 1 2 4 1 = Ñ a = 2. C\ be = / · C = — 2 4 C'\ C') + af = 2. 1 a + af)� a� 4 = 1 = 2. = V a 0 , XiY+x ..., x ...,

11 x - 6 , + â/, + · s + = 4�
x - 6 , + â/, + · s + = 4� — = Ô Ö 0, and let T° be an the Q° = q(T°) a of Q. Then the for the the normal ' tyP Lemma its 2 Q° that is, the mor- T° is the -^ of Q° is either the of its a not a a the ~

12 ^/ of Q° is a V. the C Q°, of Q° is a
^/ of Q° is a V. the C Q°, of Q° is a a 2 � 2. = 2 a a a a F\ R a a � = V T° proper a � 2. T° fi- è a a a T^ 2 a a = Q a a ïú C a a a Z, and that V contains a Then the contain

13 s a of the family of on V the T° V is
s a of the family of on V the T° V is q 8 only a of a a + a 2 / V. = a a a � (jr — h Ä = + — = � a � a a T° V = a Æ · — 2 4 C V. = a & V a 3 9 � a a Æ - the ^ Æ pape

14 rs. a = · — · � I a 2 2 Æ
rs. a = · — · � I a 2 2 Æ A º (g and a V a V' V Æ H* = ð ó: V = ö Ê proper / = , + = , V' V + = = = a ×' a = ~ the Æ has the Z' the one if g 5 the V' is - Z'\ on Z' the is the the f is

15 the a the R a V". has d 1 the d + Z
the a the R a V". has d 1 the d + Z (see has no g - = 2g — · so that the of 0 = Æ� 5, V cut a = , + a —* V" 1 a - Ã) Z' ~ s + af, á = 3 á = a (s +2f · Z') = 3 - 2 = s a + a R a a 3 + a a

16 Æ'), ×¹' a Ç' a (V, 2') -y H° = H' (
Æ'), ×¹' a Ç' a (V, 2') -y H° = H' (V, ö (V, X" Z9 a U U a - Z') a + I f ( ^ ö H' + = � (—Kv)^Ov (Ç· Æ') = Ïí ^ = Æ È a 1 a the conditions of Lemma using the notation assertions valid: then ti( =

17 0 i� * - � 7 \H* — 2Z'\
0 i� * - � 7 \H* — 2Z'\ on V' is without fixed components, base locus consists just of the d 1 Zf {see the notation). that g� L = \H* - 2Z'\ ç Æ' be the trace of the linear \H* - 2Z'\ on Z'. C \2

18 s + 3f\ = has no fixed components, base
s + 3f\ = has no fixed components, base points are just the d + z (and hence d � gl = + — - ft - 7 Ö 1 - = 0 1 g� 7 in addition that the d + z simple base points L; that is, that each z multiplicity L a

19 nd is resolved by a base lines Zf are a
nd is resolved by a base lines Zf are also simple base lines for \H* - 2Z'\, and if ô: V' V is of all the Zf for i = . . . , + ø V the map by the linear system [H* - 2Z' - ÀÞ=\ If the notation), then ö a morphism. the

20 conditions of h\0 - = 1 d Ö 1 then Q' ~
conditions of h\0 - = 1 d Ö 1 then Q' ~ - 3Z'. 9 the conditions of then ø æ a birational morphism possibly for the one case g = 9, d = 2) which contracts the sur- Q' (in the notation of some irreducible curve Õ and whic

21 h contracts the surfaces = some line Y
h contracts the surfaces = some line Y the conditions of have the isomorphism ^ W is nonsingular if and only if ö Æ' ø is an isomorphism. is nonsingular, then it is a Fano 3-fold of the first species and of index r�

22 00; 2 in P - -+ 0, O - -y Ov 0 (H*
00; 2 in P - -+ 0, O - -y Ov 0 (H* — 21') ~ Æ')� - Ïæ· (Ç* — Z') 0� = 0� � = 0 ß 2Z'))=g — — a a - - · proper a C Q. a C V, = Æ Æ - 2Z'\ a = = 5� - 2Z'\ Å V' - (J f'

23 =é Zf find, a G H'. 0 = - a a - 2Z'\
=é Zf find, a G H'. 0 = - a a - 2Z'\, \Jf=l — 2Z'\ + a - - 2Z') = \2s + s / ^ + - 2Z'\ + W) 2Z') = 0� 2. = + d 1 - 2Z'\ ~ 2Z'\ 1 + 1 - = + ^ a - a - 2Z'\ 1 � 0 = I tf* — — 2 · • Z

24 fi - · - 2 — 4 + 2 = = 0 = , Ð = 1 -
fi - · - 2 — 4 + 2 = = 0 = , Ð = 1 - + a V = - = 1 - = C proper - Æ - a Æ · ~ H* - = · - a + aZ' - á - 2Z') C V 0 - 1Z' - — 2 = 0. • \H* — Öº - - - 22 + d+l. 1 + . � 4 a 5 = a Ö0 Ð Æ' 3

25 Æ' i = , ^ Z. = - 2Z') ~ aH* - bZ''
Æ' i = , ^ Z. = - 2Z') ~ aH* - bZ''. a Æ È · - 2Z') - Æ', 2Z')— Q'. - a · (¹* - 2Z' - a d = a a a a Æ' a a C W. = Q' 1 a a — 2Z\ W � J a a a R R 1 1 V Fano 3-fold of the first species.

26 that V contains a line, and let ð V the
that V contains a line, and let ð V the double projection {see a sufficiently general (in the sense of Lemma Æ Let Å denote the section of following assertions hold: g W - W Fano 3-fold of the first species and of index

27 2 possibly one singular point); the ma
2 possibly one singular point); the map ñ W inverse to n given linear system \3E with Õ a normal rational curve of degree do not exist any Fano 3-folds of the first species with g = W a quadric and ñ W is given by the

28 linear \5E where Õ is a smooth curve of
linear \5E where Õ is a smooth curve of genus 2 and degree 1 in then W = ñ V is given by the linear system \1E Õ is a smooth curve of genus 3 and degree g ð V a rational map with fibers (after resolving the de- of genu

29 s 2, and such that the inverse images of
s 2, and such that the inverse images of lines ofP rational g ð a rational map whose general fiber (after resolv- the indeterminacy) is a del Pezzo surface of degree blown up; V is a ra- the projection from a line maps i

30 t into a complete intersection of 3 quad
t into a complete intersection of 3 quad- of a smooth rational ruled surface R 3-folds of the first species with g 7 g� 9 are course, assuming that there exist lines on them first 8 = a = = a � = = -

31 3Z')) = 3 a first a = W a V' ^ =
3Z')) = 3 a first a = W a V' ^ = Æ · Ö A a 1 = a a 1 7 1 = = 2 = 3 fiber fiber a - 2Z'\, Æ 3 3 a s + 3 a a find — — 3Z' find + + = — = z G \H* — 2Z'\ a Ã) H' = r + r^Z\ + X, / = · Zf = - + ·

32 · Zf) 2 / = a 2 = = Æ a Y' fiber
· Zf) 2 / = a 2 = = Æ a Y' fiber a a ilZ', m a W a a / = t 2 - \ _ 1 3 a find V = nE—mY — ^ m , m, = {C C C 1 U ö C W a 5 3 a a a 5 a 4 , a 3 a a = a a Y + a = a W — Y' — Õ Y^— Y'

33 *� \\ — — = a a 1 a 3 ^ a
*� \\ — — = a a 1 a 3 ^ a —y, Õ'Ë 2 , a = 2 = a = = Q' ~ H* - , 2 Æ · a — Æ · = the a rational map ñ W the linear - Wq a two 7 = is a Æ is a and the Y and Z. and Y , a = 1 ... , z = a =

34 C\ Æ · a ~ IE = ~ + If 4 a + · Zf
C\ Æ · a ~ IE = ~ + If 4 a + · Zf - - · = · = - + (X · Z?)� , 3 · · · = - 3Z', C a 7 = + · + = = W 1 2 3 = = Æ · Ç. = Q' ... , z a fi- = - = V' — 2Æ'— J = G - a — Ç' Ç' = Ç' ~ + 0. =

35 H a fiber ^ H' = + (X = º. Ê · = (D
H a fiber ^ H' = + (X = º. Ê · = (D - = - - ' - | 8 a K^,)^, a a 5 + 0 = - · H* = (X' + 3f) Ö a 5 a a a a 1 = 8 , 5 , + 8 a V' a a a 3 a 3 a ... , x X x ... , x / = 3 / = 8 V 2 C a a é 3

36 C a C a a a cui puo due 8 tre r
C a C a a a cui puo due 8 tre ricerche sulle varieta algebriche a tre dimensioni a curve-sezioni canoniche, de construction et theoremes d'existence en geometrie 3-folds. , varieties of the first genus, 3 theor

37 em� V^ algebriche su cui I'aggi
em� V^ algebriche su cui I'aggiunzione si estingue, 9 threefolds, with special regard to problems of rationality, models of K-3 surfaces, the quartic threefold, lectures on three-dimensional varieties, 5 the

38 intersection of quadrics, 6 to algebrai
intersection of quadrics, 6 to algebraic geometry, Noether-Enriques theorem on canonical curves, smoothness of the general anticanonical divisor on a Fano variety, Izv. existence of lines on Fano 3-folds, theory of Fan