By Kenji Ueno (auth.), Alberto Conte (eds.)
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Additional resources for Algebraic Threefolds: Proceedings of the 2nd1981 Session of the Centro Internazionale Matematico Estivo (C.I.M.E.), Held at Varenna, Italy, June 15–23, 1981
6. EX~I~LES OF FANO VARIETIES OF DIMENSION 3 (= FANO 3-FOLDS) In the following H denotes always a hyperplane i ) V = ~ 3 ; KV = -4H . ii ) V = V(2) = ~ 4 ; KV = -3H . section. iii ) V = V(3) a ~ 4 ; KV = -2H . iv ) V = V(4) a ~ 4 = Kv = -H . v ) V = V(3,2) = m 5 ; K V = -H . H viii) Let X = ]P7 ; KV = -H . ) in ]P8 ' take . u c ]P8 ; KV = -H . Note that in the examples i),-,vi) we have vii) and viii) Pic(V) -~ 7z ~ 7z . Pic(V) -~ 2Z, in examples 40 l9 7. G E N E R A L These PROPERTIES general a) Serre d u a l i t y n then hi(D) themrem.
As we shall see in the next section, is not necessarily invariant the Kodaira dimensio under smooth deformations in the analytic category. 2. Let the Albanese mapping g1(v) ~ V threefolds be an algebraic ~ : V---~A(V) with threefold with is surjective. ~=0. Then Hence we have 3. I) If g1(V)= 3, then the Albanese mapping 2) If g1(V) = 2, then an 6tale fibre bundle over In t h i s k = 0. case we have ~ : V ~ - ~ A(V) ~ is birational. is birationally equivalent A(V) whose fibre is an elliptic pg(V)-~ 0 .
Look to the exponential sequence 0 --§ ~ --+ exp r) --+ O* --+ h I (0 V) = h 2 (0 V) = 0 . and use COROLLARY Kv. C2(V) 5. = -24 i PROOF. Apply Riemann-Roch LE~A to D = 0 and use Corollary 2. 2. h0(_mK) m(m+l)(2m+]) 12 (_K 3) + 2m + I (me0) In paticular h0(-K) = -~K 3 + 3 . PROOF. 8. in~nediately by applying Riemann-Roch to D . X E ]-Kvt CONSEOUE~CES FOR Assume X c I-KvI that By the adjunction is smooth (cf. I0). = -mm and using 42 0 and the fact that ---+ :)V(-x]" " ---~ :~V ---+ C)X :=> HI(Ov) = H2(0v(-X)) = 0 (Lemma ]) we get q(X) = dim H ](X,r~X) = 0 .