Geography on 3-folds of General Type. September 9, Meng Chen Fudan University, Shanghai
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1 Meng Chen Fudan University, Shanghai September 9, 2010
2 A Classical Problem Let V be a nonsingular projective variety. Consider the canonical line bundle ω V = O V (K V ). One task of birational geometry is to study the geometry induced from linear system mk V or mk V, m Z +.
3 A Classical Problem Let V be a nonsingular projective variety. Consider the canonical line bundle ω V = O V (K V ). One task of birational geometry is to study the geometry induced from linear system mk V or mk V, m Z +. Assume that V is of general type, i.e. κ(v ) = dim(v ). Set V n := {n-dimensional variety of general type}.
4 A Classical Problem Let V be a nonsingular projective variety. Consider the canonical line bundle ω V = O V (K V ). One task of birational geometry is to study the geometry induced from linear system mk V or mk V, m Z +. Assume that V is of general type, i.e. κ(v ) = dim(v ). Set V n := {n-dimensional variety of general type}. Post-MMP Problem: how to classify V n?
5 Pluricanonical boundedness In 2006, Hacon-M c kernan, Takayama r n such that ϕ m is birational m r n and V V n.
6 Pluricanonical boundedness In 2006, Hacon-M c kernan, Takayama r n such that ϕ m is birational m r n and V V n. Chen-Chen (1) r 3 73; (2) Vol(V ) 1/2660 V V 3.
7 Pluricanonical boundedness In 2006, Hacon-M c kernan, Takayama r n such that ϕ m is birational m r n and V V n. Chen-Chen (1) r 3 73; (2) Vol(V ) 1/2660 V V 3. The aim of this talk geography to improve the above results.
8 Geography Let X be a (QFT) minimal projective 3-fold of general type. Reid! weighted basket B X := {B X, P 2, O X } such that all the birational invariants of X are uniquely determined by B X, where B X = { 1 r i (1, 1, b i ) i = 1,..., t}.
9 Geography Let X be a (QFT) minimal projective 3-fold of general type. Reid! weighted basket B X := {B X, P 2, O X } such that all the birational invariants of X are uniquely determined by B X, where B X = { 1 r i (1, 1, b i ) i = 1,..., t}. Open problem: to find exact relations between the sets V 3 {weighted baskets}
10 Two geographical inequalities Miyaoka-Reid inequality: K 3 X 72χ(ω X ) + 3 i (r i 1 r i ).
11 Two geographical inequalities Miyaoka-Reid inequality: K 3 X 72χ(ω X ) + 3 i (r i 1 r i ). Inequalities of Noether type (Chen-Chen): K 3 X a m P m (X ) b m where a m, b m Q +, m 1.
12 Numerical genus The fact: general type 3-folds with p g 1 form an infinite family.
13 Numerical genus The fact: general type 3-folds with p g 1 form an infinite family. When p g (X ) 1, n 0 (X ) := min{m P m (X ) 2}. Chen-Chen 2 n 0 (X ) 18. Definition The numerical genus of X is defined as: { pg (X ); p g (X ) 2 g(x ) := otherwise. 1 n 0 (X ) ;
14 The Noether function N (g) Chen-Chen g(x ) 1 18.
15 The Noether function N (g) Chen-Chen g(x ) The Noether function N (g) := inf{kx 3 g(x ) = g}.
16 The Noether function N (g) Chen-Chen g(x ) The Noether function N (g) := inf{kx 3 g(x ) = g}. For all minimal 3-fold X of general type, Noether inequality K 3 X N (g(x )).
17 The Noether function N (g) Chen-Chen g(x ) The Noether function N (g) := inf{kx 3 g(x ) = g}. For all minimal 3-fold X of general type, Noether inequality K 3 X N (g(x )). What is the Noether function N (g)?
18 Noether inequalities in narrow sense In 1992, Kobayashi constructed a family of canonically polarized 3-folds satisfying: K 3 X = 4 3 p g(x ) 10 3.
19 Noether inequalities in narrow sense In 1992, Kobayashi constructed a family of canonically polarized 3-folds satisfying: KX 3 = 4 3 p g(x ) In 2004, Chen KX p g(x ) 10 3 for canonically polarized 3-folds.
20 Noether inequalities in narrow sense In 1992, Kobayashi constructed a family of canonically polarized 3-folds satisfying: KX 3 = 4 3 p g(x ) In 2004, Chen KX p g(x ) 10 3 for canonically polarized 3-folds. In 2006, Catanese-Chen-Zhang KX p g(x ) 10 3 for nonsingular minimal 3-folds of general type.
21 Noether inequalities in narrow sense In 1992, Kobayashi constructed a family of canonically polarized 3-folds satisfying: KX 3 = 4 3 p g(x ) In 2004, Chen KX p g(x ) 10 3 for canonically polarized 3-folds. In 2006, Catanese-Chen-Zhang KX p g(x ) 10 3 for nonsingular minimal 3-folds of general type. Conjecture: KX p g(x ) 10 3 holds for Gorenstein minimal 3-folds of general type.
22 Known value of N (g) In 2007, Chen K 3 X 1 3 when g = p g (X ) 2.
23 Known value of N (g) In 2007, Chen K 3 X 1 3 when g = p g (X ) 2. Chen N (2) = 1 3 ; N (3) = 1; N (4) = 2; N (g) g 2 for g 5. due to supporting examples of Fletcher-Reid.
24 The strategy to get the lower bound of K 3 X Fletcher-Reid s example: X 46 P(4, 5, 6, 7, 23), K 3 =
25 The strategy to get the lower bound of K 3 X Fletcher-Reid s example: X 46 P(4, 5, 6, 7, 23), K 3 = When p g (X ) 1, 1 18 g 1 2.
26 The strategy to get the lower bound of K 3 X Fletcher-Reid s example: X 46 P(4, 5, 6, 7, 23), K 3 = When p g (X ) 1, 18 g 1 2. When KX 3 < 1 420, Reid s weighted baskets can be completely listed, but the list is too big!
27 The strategy to get the lower bound of K 3 X Fletcher-Reid s example: X 46 P(4, 5, 6, 7, 23), K 3 = When p g (X ) 1, 18 g 1 2. When KX 3 < 1 420, Reid s weighted baskets can be completely listed, but the list is too big! To find a function c(g) such that K 3 X c(g) with g(x ) = g.
28 The main statements Chen-Chen a very effective function v(g) (g < 2) satisfying KX 3 v(g(x )).
29 The main statements Chen-Chen a very effective function v(g) (g < 2) satisfying KX 3 v(g(x )). Set g = 1/n 0, here is part of the description: n v(n 0 ) 1/420 1/450 1/630 1/825 1/1089 1/1404 n v(n 0 ) 1/1728 1/ /2640
30 The main statements Chen-Chen a very effective function v(g) (g < 2) satisfying KX 3 v(g(x )). Set g = 1/n 0, here is part of the description: n v(n 0 ) 1/420 1/450 1/630 1/825 1/1089 1/1404 n v(n 0 ) 1/1728 1/ /2640 N ( 1 2 ) = v(1 2 ) = (optimal)
31 Conclusions Fletcher-Reid examples with g = 1/2 and K 3 = 1/12: X 22 P(1, 2, 3, 4, 11) X 6,18 P(2, 2, 3, 3, 4, 9) X 10,14 P(2, 2, 3, 4, 5, 7)
32 Conclusions Fletcher-Reid examples with g = 1/2 and K 3 = 1/12: X 22 P(1, 2, 3, 4, 11) X 6,18 P(2, 2, 3, 3, 4, 9) X 10,14 P(2, 2, 3, 4, 5, 7) Theorem Let X be a minimal projective 3-fold of general type. Then (1) KX > Furthermore, KX 3 = if and only if B(X ) = {B 3a, 0, 3}. (2) (announcement) ϕ m is birational for m 65.
33 The method We study the m 0 -canonical map of X : ϕ m0 : X P P m 0 1. By Hironaka s big theorem, we can take successive blow-ups π : X X such that: (i) X is smooth; (ii) the movable part of m 0 K X is base point free; (iii) the support of the union of π (K m0 ) and the exceptional divisors is of simple normal crossings.
34 The method Set g m0 := ϕ m0 π. Then g m0 is a morphism by assumption. Let X f Γ s W be the Stein factorization of g m0 with W the image of X through g m0. X f Γ π g m0 X ϕ m0 s W
35 The method Denote by M m0 the movable part of m 0 K X. One has m 0 π (K X ) = M m0 + E m 0 for an effective Q-divisor E m 0. In total, since h 0 (X, m 0 π (K X ) ) = h 0 (X, m 0 π (K X ) ) = P m0 (X ) = P m0 (X ), one has: m 0 K X = M m0 + Z m0 where Z m0 is the fixed part of m 0 K X.
36 The method If dim(γ) 2, a general member S of M m0 is a nonsingular projective surface of general type. Set p = 1.
37 The method If dim(γ) 2, a general member S of M m0 is a nonsingular projective surface of general type. Set p = 1. If dim(γ) = 1, a general fiber S of f is an irreducible smooth projective surface of general type. We may write a m0 M m0 = S i a m0 S i=1 where S i are smooth fibers of f for all i and a m0 min{2p m0 2, P m0 + g(γ) 1}. Set p = a m0.
38 The method Let S be a generic irreducible element of m 0 K X. Let G be a base point free linear system on S. Let C be a generic irreducible element of G. Kodaira Lemma β > 0 such that π (K X ) S βc.
39 The method Let S be a generic irreducible element of m 0 K X. Let G be a base point free linear system on S. Let C be a generic irreducible element of G. Kodaira Lemma β > 0 such that π (K X ) S βc. Inequality (1): where ξ = π (K X ) C. K 3 X pβ m 0 ξ (1)
40 The method Inequality (2): ξ deg(k C) 1 + m 0 p + 1 β. (2)
41 The method Inequality (2): ξ deg(k C) 1 + m 0 p + 1 β. (2) Inequality (3): For any positive integer m such that α m := (m 1 m 0 p 1 β )ξ > 1, one has ξ deg(k C) + α m. (3) m
42 Technical applications When dim Γ > 1, take G := S S. Thus β = 1 m 0.
43 Technical applications When dim Γ > 1, take G := S S. Thus β = 1 m 0. When dim Γ = 1, take G = qσ (K S0 ) for q 1 where σ : S S 0 is the contraction onto the minima model. Here is a key inequality: π (K X ) S p m 0 + p σ (K S0 ).
44 Technical applications When dim Γ > 1, take G := S S. Thus β = 1 m 0. When dim Γ = 1, take G = qσ (K S0 ) for q 1 where σ : S S 0 is the contraction onto the minima model. Here is a key inequality: π (K X ) S p m 0 + p σ (K S0 ). Here is the complete list for 3-folds with small invariants:
45 No. (P 3,, P 11 ) P 18 P 24 µ 1 χ B (12) = (n 1,2, n 5,11,, n 1,5 ) or B min K 3 1 (0, 0, 0, 0, 0, 0, 0, 1, 0) (5, 0, 0, 1, 0, 3, 0, 0, 3, 0, 0, 1, 0, 0, 0) (0, 0, 0, 0, 0, 1, 0, 0, 0) (4, 0, 1, 0, 0, 2, 1, 0, 3, 0, 0, 0, 2, 0, 0) a {(2, 5), (3, 8), } {(5, 13), } (0, 0, 0, 0, 0, 1, 0, 1, 0) (6, 1, 0, 0, 0, 4, 1, 0, 4, 0, 1, 0, 2, 0, 0) a {(2, 5), (3, 8), } {(5, 13), } (0, 0, 0, 0, 0, 1, 0, 1, 0) (7, 0, 1, 0, 0, 4, 0, 1, 3, 0, 1, 0, 2, 0, 0) {(4, 11), (1, 3), } {(5, 14), } (0, 0, 0, 0, 0, 1, 0, 1, 0) (7, 0, 1, 0, 0, 4, 1, 0, 4, 0, 0, 1, 1, 0, 0) a {(8, 20), (3, 8), } {(11, 28), } b {(5, 13), (4, 15), } (0, 0, 0, 1, 0, 0, 0, 1, 0) (9, 0, 0, 2, 0, 1, 0, 1, 4, 0, 2, 0, 0, 0, 1) (0, 0, 0, 1, 0, 0, 1, 0, 0) (5, 0, 1, 1, 0, 0, 0, 0, 5, 0, 1, 0, 0, 0, 1) a {(4, 9), (3, 7), } {(7, 16), } (0, 0, 0, 1, 0, 0, 1, 1, 0) (7, 1, 0, 1, 0, 2, 0, 0, 6, 0, 2, 0, 0, 0, 1) (0, 0, 0, 1, 0, 1, 0, 0, 0) (8, 0, 1, 1, 0, 0, 2, 0, 5, 0, 1, 0, 1, 0, 1) a {(4, 9), (3, 7), } {(7, 16), } (0, 0, 0, 1, 0, 1, 0, 1, 0) (9, 0, 0, 2, 0, 0, 1, 1, 3, 1, 0, 0, 1, 0, 1) (0, 0, 0, 1, 0, 1, 0, 1, 0) (9, 0, 1, 0, 0, 1, 2, 0, 4, 0, 2, 0, 0, 0, 1) a {(2, 5), (6, 16), } {(8, 21), } (0, 0, 0, 1, 0, 1, 0, 1, 0) (12, 0, 0, 2, 0, 2, 0, 2, 4, 0, 2, 0, 0, 1, 0) (0, 0, 0, 1, 0, 1, 0, 1, 0) (10, 1, 0, 1, 0, 2, 2, 0, 6, 0, 2, 0, 1, 0, 1) (0, 0, 0, 1, 0, 1, 0, 1, 0) (11, 0, 1, 1, 0, 2, 1, 1, 5, 0, 2, 0, 1, 0, 1) b {(2, 5), (3, 8), } {(5, 13), } c {(7, 16), (7, 19), } (0, 0, 0, 1, 0, 1, 0, 1, 0) (11, 0, 1, 1, 0, 2, 2, 0, 6, 0, 1, 1, 0, 0, 1) {(2, 5), (3, 8), } {(5, 13), }
46 16b {(2, 5), (6, 16), } {(8, 21), } {(4, 9), (3, 7), } {(7, 16), } (0, 0, 0, 1, 0, 1, 0, 1, 1) (9, 0, 0, 2, 0, 0, 0, 2, 3, 0, 1, 0, 1, 0, 1) (0, 0, 0, 1, 0, 1, 0, 1, 1) (9, 0, 0, 2, 0, 0, 1, 1, 4, 0, 0, 1, 0, 0, 1) b {(3, 8), (4, 11), } {(7, 19), } (0, 0, 0, 1, 0, 1, 1, 0, 0) (8, 0, 1, 1, 0, 1, 0, 1, 5, 0, 1, 0, 0, 1, 0) (0, 0, 0, 1, 0, 1, 1, 0, 0) (7, 0, 2, 0, 0, 1, 1, 0, 6, 0, 1, 0, 1, 0, 1) (0, 0, 0, 1, 0, 1, 1, 1, 0) (6, 0, 1, 0, 0, 0, 1, 0, 3, 1, 0, 0, 0, 0, 1) (0, 0, 0, 1, 0, 1, 1, 1, 0) (8, 0, 1, 1, 0, 1, 0, 1, 4, 1, 0, 0, 1, 0, 1) (0, 0, 0, 1, 0, 1, 1, 1, 0) (9, 1, 1, 0, 0, 3, 1, 0, 7, 0, 2, 0, 1, 0, 1) a {(5, 11), (4, 9), } {(9, 20), } (0, 0, 0, 1, 0, 1, 1, 1, 0, ) (10, 0, 2, 0, 0, 3, 0, 1, 6, 0, 2, 0, 1, 0, 1) a {(4, 11), (1, 3), } {(5, 14), } (0, 0, 0, 1, 0, 1, 1, 1, 0) (10, 0, 2, 0, 0, 3, 1, 0, 7, 0, 1, 1, 0, 0, 1) {(4, 10), (3, 8), } {(7, 18), } b {(5, 13), (5, 18), } (0, 0, 0, 1, 0, 1, 1, 1, 1) (5, 1, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 1) (0, 0, 0, 1, 0, 1, 1, 1, 1) (6, 0, 1, 0, 0, 0, 0, 1, 3, 0, 1, 0, 0, 0, 1) {(4, 11), (1, 3), } {(5, 14), } (0, 0, 0, 1, 0, 1, 1, 1, 1) (7, 1, 0, 1, 0, 1, 0, 1, 5, 0, 1, 0, 1, 0, 1) (0, 0, 0, 1, 0, 1, 1, 1, 1) (7, 1, 0, 1, 0, 1, 1, 0, 6, 0, 0, 1, 0, 0, 1) (0, 0, 0, 1, 0, 1, 1, 1, 1) (8, 0, 1, 1, 0, 1, 0, 1, 5, 0, 0, 1, 0, 0, 1) a {(4, 9), (3, 7), } {(7, 16), } b {(4, 11), (1, 3), } {(5, 14), } (0, 0, 0, 1, 1, 0, 0, 1, 0) (5, 0, 0, 2, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0) (0, 0, 0, 1, 1, 0, 0, 1, 0) (7, 0, 1, 1, 0, 2, 1, 0, 3, 0, 3, 0, 0, 0, 0) a {(4, 9), (3, 7), } {(7, 16), } b {(2, 5), (3, 8), } {(5, 13), }
47 No. (P 3,, P 11 ) P 18 P 24 µ 1 χ (n 1,2, n 4,9,, n 1,5 ) or B min K 3 35 (0, 0, 0, 1, 1, 0, 0, 1, 1) (5, 0, 0, 2, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0) (0, 0, 0, 1, 1, 0, 1, 1, 0) (4, 0, 1, 1, 0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 0) a {(4, 9), (3, 7), } {(7, 16), } b {(3, 10), (2, 7), } {(5, 17), } (0, 0, 0, 1, 1, 0, 1, 1, 0) (6, 0, 2, 0, 0, 3, 0, 0, 4, 0, 3, 0, 0, 0, 0) (0, 0, 0, 1, 1, 0, 1, 1, 1) (3, 1, 0, 1, 0, 1, 0, 0, 3, 0, 2, 0, 0, 0, 0) (0, 0, 0, 1, 1, 1, 0, 1, 0) (7, 0, 1, 1, 0, 1, 2, 0, 2, 1, 1, 0, 1, 0, 0) a {(4, 9), (3, 7), } {(7, 16), } b {(3, 10), (2, 7), } {(5, 17), } (0, 0, 0, 1, 1, 1, 0, 1, 0) (9, 0, 2, 0, 0, 3, 2, 0, 4, 0, 3, 0, 1, 0, 0) {(2, 5), (3, 8), } {(5, 13), } > b {(2, 5), (6, 16), } {(8, 21), } (0, 0, 0, 1, 1, 1, 0, 1, 1) (5, 0, 1, 0, 0, 0, 2, 0, 1, 0, 2, 0, 0, 0, 0) (0, 0, 0, 1, 1, 1, 0, 1, 1) (6, 1, 0, 1, 0, 1, 2, 0, 3, 0, 2, 0, 1, 0, 0) (0, 0, 0, 1, 1, 1, 0, 1, 1) (7, 0, 1, 1, 0, 1, 1, 1, 2, 0, 2, 0, 1, 0, 0) b {(2, 5), (3, 8), } {(5, 13), } c {(7, 16), (7, 19), } (0, 0, 0, 1, 1, 1, 0, 1, 1) (7, 0, 1, 1, 0, 1, 2, 0, 3, 0, 1, 1, 0, 0, 0) a {(2, 5), (6, 16), } {(8, 21), } c {(7, 16), (5, 18), } {(5, 13), } > (0, 0, 0, 1, 1, 1, 1, 0, 1) (3, 0, 2, 0, 0, 0, 1, 0, 3, 0, 1, 0, 1, 0, 0) (0, 0, 0, 1, 1, 1, 1, 1, 0) (6, 0, 2, 0, 0, 2, 1, 0, 3, 1, 1, 0, 1, 0, 0) b {(3, 10), (2, 7), } {(5, 17), } , 0, 0, 1, 1, 1, 1, 1, 1) (4, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0) (0, 0, 0, 1, 1, 1, 1, 1, 1) (5, 1, 1, 0, 0, 2, 1, 0, 4, 0, 2, 0, 1, 0, 0)
48 49a {(5, 11), (4, 9), } {(9, 20), } (0, 0, 0, 1, 1, 1, 1, 1, 1) (6, 0, 2, 0, 0, 2, 0, 1, 3, 0, 2, 0, 1, 0, 0) a {(4, 11), (1, 3), } {(5, 14), } (0, 0, 0, 1, 1, 1, 1, 1, 1) (6, 0, 2, 0, 0, 2, 1, 0, 4, 0, 1, 1, 0, 0, 0) a {(4, 10), (3, 8), } {(7, 18), } b {(5, 13), (5, 18), } (0, 0, 1, 0, 0, 1, 0, 1, 0) (4, 0, 0, 1, 0, 2, 2, 0, 2, 0, 0, 0, 0, 0, 1) (0, 0, 1, 0, 0, 1, 1, 1, 0) (3, 0, 1, 0, 0, 3, 1, 0, 3, 0, 0, 0, 0, 0, 1) a {(2, 5), (3, 8), } {(5, 13), } (0, 0, 1, 0, 1, 0, 0, 1, 0) (2, 0, 0, 2, 0, 3, 1, 0, 1, 0, 1, 0, 0, 0, 0) (0, 0, 1, 0, 1, 0, 1, 1, 0) (1, 0, 1, 1, 0, 4, 0, 0, 2, 0, 1, 0, 0, 0, 0) a {(4, 9), (3, 7), } {(7, 16), } (0, 0, 1, 0, 1, 0, 1, 1, 0) (3, 0, 1, 2, 0, 5, 0, 0, 4, 0, 0, 1, 0, 0, 0) (0, 0, 1, 0, 1, 1, 0, 1, 0) (4, 0, 1, 1, 0, 4, 2, 0, 2, 0, 1, 0, 1, 0, 0) a {(4, 9), (3, 7), } {(7, 16), } (0, 0, 1, 0, 1, 1, 0, 1, 1) (2, 0, 0, 2, 0, 2, 1, 1, 0, 0, 0, 0, 1, 0, 0) (0, 0, 1, 0, 1, 1, 1, 1, 0) (3, 0, 2, 0, 0, 5, 1, 0, 3, 0, 1, 0, 1, 0, 0) (0, 0, 1, 0, 1, 1, 1, 1, 1) (1, 0, 1, 1, 0, 3, 0, 1, 1, 0, 0, 0, 1, 0, 0)
49 Canonically fibred 3-folds Let X be a nonsingular projective 3-fold of general type. When the geometric genus p g 2, the canonical map ϕ 1 := Φ KX is usually a key tool for birational classification.
50 Canonically fibred 3-folds Let X be a nonsingular projective 3-fold of general type. When the geometric genus p g 2, the canonical map ϕ 1 := Φ KX is usually a key tool for birational classification. If ϕ 1 is of fiber type (i.e. dim ϕ 1 (X ) < 3), it is interesting to see if the birational invariants of the generic irreducible component in the general fiber of ϕ 1 is bounded from above.
51 Canonically fibred 3-folds Let X be a nonsingular projective 3-fold of general type. When the geometric genus p g 2, the canonical map ϕ 1 := Φ KX is usually a key tool for birational classification. If ϕ 1 is of fiber type (i.e. dim ϕ 1 (X ) < 3), it is interesting to see if the birational invariants of the generic irreducible component in the general fiber of ϕ 1 is bounded from above. Chen-Hacon When X is Gorenstein minimal and ϕ 1 is of fiber type, then X is canonically fibred by surfaces or curves with bounded invariants.
52 Canonically fibred 3-folds Chen-Cui, 2010 Theorem Let X be a Gorenstein minimal projective 3-fold of general type. Assume that X is canonically of fiber type. Let F be a smooth model of the generic irreducible component in the general fiber of ϕ 1. Then (i) g(f ) 91 when F is a curve and p g (X ) 183; (ii) p g (F ) 37 when F is a surface and p g (X ) 0, say p g (X ) 3890.
53 New examples Standard construction. Let S be a minimalsurface of g eneral type with p g (S) = 0. Assume there exists a divisor H on S such that K S + H is composed with a pencil of curves and that 2H is linearly equivalent to a smooth divisor R. Let Ĉ be a generic irreducible element of the movable part of K S + H. Assume Ĉ is smooth. Set d := Ĉ.H and D := Ĉ H. Let C 0 be a fixed smooth projective curve of genus 2. Let θ be a 2-torsion divisor on C 0. Set Y := S C 0. Take δ := p1 (H) + p 2 (θ) and pick a smooth divisor p1 (2H). Then the pair (δ, ) determines a smooth double covering π : X Y and K X = π (K Y + δ).
54 New examples Since K Y + δ = p1 (K S + H) + p2 (K C 0 + θ), p g (Y ) = 0 and h 0 (K C0 + θ) = 1, one sees that K X = π K Y + δ and that Φ KX factors through π, p 1 and Φ KS +H. Since K S + H is composed with a pencil of curves Ĉ, X is canonically fibred by surfaces F and F is a double covering over T := Ĉ C 0 corresponding to the data (q1 (D) + q 2 (θ), q 1 (2D)) where q 1 and q 2 are projections. Denote by σ : F T the double covering. Then K F = σ (K T + q1 (D) + q 2 (θ)). By calculation, one has p g (F ) = 3g(Ĉ) when d = 0 and p g (F ) = 3g(Ĉ) + d 1 whenever d > 0.
55 New examples Lemma Let S be any smooth minimal projective surface of general type with p g (S) = 0. Assume µ : S P 1 is a genus 2 fibration. Let H be a general fiber of µ. Then K S + H is composed with a pencil of curves Ĉ of genus g(ĉ) and Ĉ.H = 2.
56 New examples Lemma Let S be any smooth minimal projective surface of general type with p g (S) = 0. Assume µ : S P 1 is a genus 2 fibration. Let H be a general fiber of µ. Then K S + H is composed with a pencil of curves Ĉ of genus g(ĉ) and Ĉ.H = 2. We take a pair (S, H) which was found by Xiao, where S is a numerical Compedelli surface with K 2 S = 2, p g(s) = q(s) = 0 and Tor(S) = (Z 2 ) 3.
57 X S,19 Let P = P 1 P 1. Take four curves C 1, C 2, C 3 and C 4 defined by the following equations, respectively: C 1 : x = y; C 2 : x = y; C 3 : xy = 1; C 4 : xy = 1. These four curves intersect mutually at 12 ordinary double points: (0, 0), (, ), (0, ), (, 0) (±1, ±1), (± 1, ± 1).
58 (0, ) (1, 1) (1, 1) ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1) (, ) (0, 0) (, 0) Q 0 P 0 P 1 P 1 P 1 P 1 P Q Q 1 Q 1 Q 1 Q 1
59 X S,19 Xiao There exists a divisor R 1 of bidegree (14, 6) which has exactly 12 simple singularities of multiplicity 4. Then the data (δ 1, R 1 ) determines a singular double covering onto P. f S σ S f θ P P 1 P 1 ϕ K 2 S = 2 and p g(s) = q(s) = 0. τ P
60 X S,19 Let H be a general fiber of f. Calculations K S + H has exactly 6 base points, but no fixed parts. Clearly a general member Ĉ K S + H is a smooth curve of genus 6.
61 X S,19 Let H be a general fiber of f. Calculations K S + H has exactly 6 base points, but no fixed parts. Clearly a general member Ĉ K S + H is a smooth curve of genus 6. Now we take the triple (S, H, Ĉ) and run standard construction. What we get is the 3-fold X S,19 which is canonically fibred by surfaces F with p g (F ) = 19.
62 X S,19 Let H be a general fiber of f. Calculations K S + H has exactly 6 base points, but no fixed parts. Clearly a general member Ĉ K S + H is a smooth curve of genus 6. Now we take the triple (S, H, Ĉ) and run standard construction. What we get is the 3-fold X S,19 which is canonically fibred by surfaces F with p g (F ) = 19. Thanks very much!
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