Geography on 3-folds of General Type. September 9, Meng Chen Fudan University, Shanghai

Size: px

Start display at page:

Download "Geography on 3-folds of General Type. September 9, Meng Chen Fudan University, Shanghai"

Kelly Conley
5 years ago
Views:

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!

The Noether Inequality for Algebraic Threefolds

The Noether Inequality for Algebraic Threefolds .... Meng Chen School of Mathematical Sciences, Fudan University 2018.09.08 P1. Surface geography Back ground Two famous inequalities c 2 1 Miyaoka-Yau inequality Noether inequality O χ(o) P1. Surface