Numeri di Chern fra topologia e geometria birazionale

Size: px
Start display at page:

Download "Numeri di Chern fra topologia e geometria birazionale"

Transcription

1 Giornate di Geometria Algebrica ed Argomenti Correlati XII (Torino, 2014) Numeri di Chern fra topologia e geometria birazionale (Work in progress with P. Cascini) Luca Tasin (Max Planck Institute for Mathematics) Luca Tasin (MPIM) Chern numbers of 3-folds 1 / 19

2 Set-up Introduction X smooth projective variety of dimension n over C. c i H 2i (X, Z) the Chern classes of the tangent bundle T X. Luca Tasin (MPIM) Chern numbers of 3-folds 2 / 19

3 Set-up Introduction X smooth projective variety of dimension n over C. c i H 2i (X, Z) the Chern classes of the tangent bundle T X. c 1 = K X H 2 (X, Z), c n Z is the Euler characteristic of X. The product of Chern classes of total degree 2n are called Chern numbers. Luca Tasin (MPIM) Chern numbers of 3-folds 2 / 19

4 Set-up Introduction X smooth projective variety of dimension n over C. c i H 2i (X, Z) the Chern classes of the tangent bundle T X. c 1 = K X H 2 (X, Z), c n Z is the Euler characteristic of X. The product of Chern classes of total degree 2n are called Chern numbers. Hirzebruch (1954) Which linear combinations of Chern numbers are topologically invariant? Luca Tasin (MPIM) Chern numbers of 3-folds 2 / 19

5 The main problem Theorem (Kotshick, 2012) A rational linear combination of Chern numbers is a topological invariant iff it is a multiple of the Euler characteristic c n. Luca Tasin (MPIM) Chern numbers of 3-folds 3 / 19

6 The main problem Theorem (Kotshick, 2012) A rational linear combination of Chern numbers is a topological invariant iff it is a multiple of the Euler characteristic c n. dim X = 2. c 2 1 has two possible values. Luca Tasin (MPIM) Chern numbers of 3-folds 3 / 19

7 The main problem Theorem (Kotshick, 2012) A rational linear combination of Chern numbers is a topological invariant iff it is a multiple of the Euler characteristic c n. dim X = 2. c1 2 has two possible values. dim X = 3. By Hirzebruch-Riemann-Roch we have 1 24 c 1c 2 = χ(o X ) = 1 h 1,0 +h 2,0 h 3,0 1+b 1 +b 2 +b 3. Luca Tasin (MPIM) Chern numbers of 3-folds 3 / 19

8 The main problem Theorem (Kotshick, 2012) A rational linear combination of Chern numbers is a topological invariant iff it is a multiple of the Euler characteristic c n. dim X = 2. c1 2 has two possible values. dim X = 3. By Hirzebruch-Riemann-Roch we have 1 24 c 1c 2 = χ(o X ) = 1 h 1,0 +h 2,0 h 3,0 1+b 1 +b 2 +b 3. Question (Kotshick) Does KX 3 = c3 1 take only finitely many values on projective algebraic structures with the same underlying 6-manifold? Luca Tasin (MPIM) Chern numbers of 3-folds 3 / 19

9 The volume If X is a variety of dimension n, then the volume of X is defined as n!h 0 (X, mk X ) vol(x) := lim sup m + m n, and X is called of general type if vol(x) > 0. Luca Tasin (MPIM) Chern numbers of 3-folds 4 / 19

10 The volume If X is a variety of dimension n, then the volume of X is defined as n!h 0 (X, mk X ) vol(x) := lim sup m + m n, and X is called of general type if vol(x) > 0. The volume is a birational invariant, but not a topological invariant if dim X 2. Luca Tasin (MPIM) Chern numbers of 3-folds 4 / 19

11 The volume If X is a variety of dimension n, then the volume of X is defined as n!h 0 (X, mk X ) vol(x) := lim sup m + m n, and X is called of general type if vol(x) > 0. The volume is a birational invariant, but not a topological invariant if dim X 2. Theorem 1 Let X be a smooth projective 3-fold of general type. Then vol(x) 64(b 1 (X) + b 3 (X) + b 2 (X)). The volume takes finitely many values on projective algebraic structures of general type with the same underlying 6-manifold. Luca Tasin (MPIM) Chern numbers of 3-folds 4 / 19

12 MMP Introduction Let X be a smooth 3-fold of general type. Then there exists a sequence of birational maps X = X 0 X 1 X m = Y such that each X i has terminal Q-factorial singularities, Luca Tasin (MPIM) Chern numbers of 3-folds 5 / 19

13 MMP Introduction Let X be a smooth 3-fold of general type. Then there exists a sequence of birational maps X = X 0 X 1 X m = Y such that each X i has terminal Q-factorial singularities, each map φ i : X i X i+1 is either a divisorial contraction to a point or a divisorial contraction to a curve or a flip (which is an isomorphism in codimension 2) Luca Tasin (MPIM) Chern numbers of 3-folds 5 / 19

14 MMP Introduction Let X be a smooth 3-fold of general type. Then there exists a sequence of birational maps X = X 0 X 1 X m = Y such that each X i has terminal Q-factorial singularities, each map φ i : X i X i+1 is either a divisorial contraction to a point or a divisorial contraction to a curve or a flip (which is an isomorphism in codimension 2) and K Y is nef, that is K Y.C 0, for any curve C in Y. Luca Tasin (MPIM) Chern numbers of 3-folds 5 / 19

15 MMP Introduction Let X be a smooth 3-fold of general type. Then there exists a sequence of birational maps X = X 0 X 1 X m = Y such that each X i has terminal Q-factorial singularities, each map φ i : X i X i+1 is either a divisorial contraction to a point or a divisorial contraction to a curve or a flip (which is an isomorphism in codimension 2) and K Y is nef, that is K Y.C 0, for any curve C in Y. Y is called a minimal model of X. Note that vol(k Y ) = K 3 Y. Luca Tasin (MPIM) Chern numbers of 3-folds 5 / 19

16 Proof of Theorem 1 I Let X = X 0 X 1 X m = Y be an MMP for X. Luca Tasin (MPIM) Chern numbers of 3-folds 6 / 19

17 Proof of Theorem 1 I Let X = X 0 X 1 X m = Y be an MMP for X. B.M.Y. inequality for minimal model of general type (Tian-Wang 2011): ( K n 2 Y. KY 2 2n + 1 ) n c 2(Y ) 0 KY K Y.c 2 (Y ) Luca Tasin (MPIM) Chern numbers of 3-folds 6 / 19

18 Proof of Theorem 1 I Let X = X 0 X 1 X m = Y be an MMP for X. B.M.Y. inequality for minimal model of general type (Tian-Wang 2011): ( K n 2 Y. KY 2 2n + 1 ) n c 2(Y ) 0 KY K Y.c 2 (Y ) Riemann-Roch for terminal 3-folds (Kawamata, Reid): χ(y, O Y ) = 1 24 K Y.c 2 (Y ) + p B(Y ) r(p) r(p) Luca Tasin (MPIM) Chern numbers of 3-folds 6 / 19

19 Proof of Theorem 1 II Hence vol(x) = vol(y ) = KY K Y.c 2 (Y ) = 8 χ(y, O Y ) + r(p) r(p) p B(Y ) Luca Tasin (MPIM) Chern numbers of 3-folds 7 / 19

20 Proof of Theorem 1 II Hence vol(x) = vol(y ) = KY K Y.c 2 (Y ) = 8 χ(y, O Y ) + r(p) r(p) p B(Y ) χ(y, O Y ) = χ(x, O X ) b 1 (X) + b 3 (X). Luca Tasin (MPIM) Chern numbers of 3-folds 7 / 19

21 Proof of Theorem 1 II Hence vol(x) = vol(y ) = KY K Y.c 2 (Y ) = 8 χ(y, O Y ) + r(p) r(p) p B(Y ) χ(y, O Y ) = χ(x, O X ) b 1 (X) + b 3 (X). Topological bound on the singularities of Y (Cascini-Zhang, 2012): p B(Y ) r(p) 2 1 r(p) 2b 2 (X). Luca Tasin (MPIM) Chern numbers of 3-folds 7 / 19

22 The main theorem To any threefold X we can associate an integral cubic form F X Z[x 1,..., x b ], which comes from the trilinear intersection form H 2 (X, Z) H 2 (X, Z) H 2 (X, Z) Z. Denote by FX the discriminant of F X. Luca Tasin (MPIM) Chern numbers of 3-folds 8 / 19

23 The main theorem To any threefold X we can associate an integral cubic form F X Z[x 1,..., x b ], which comes from the trilinear intersection form H 2 (X, Z) H 2 (X, Z) H 2 (X, Z) Z. Denote by FX the discriminant of F X. Theorem 2 Let X be a smooth 3-fold of general type. Assume that FX 0 and that there is an MMP for X composed only by divisorial contractions to points and blow-downs to smooth curves. Then there exists a topological invariant D X such that K 3 X D X. Luca Tasin (MPIM) Chern numbers of 3-folds 8 / 19

24 Strategy Introduction Let X = X 0 X 1 X m = Y be an MMP as in Theorem 2. Luca Tasin (MPIM) Chern numbers of 3-folds 9 / 19

25 Strategy Introduction Let X = X 0 X 1 X m = Y be an MMP as in Theorem 2. K 3 Y 64(b 1(X) + b 3 (X) + b 2 (X)). At each step we want to bound K 3 X i K 3 X i+1 with a topological invariant of X. Luca Tasin (MPIM) Chern numbers of 3-folds 9 / 19

26 Strategy Introduction Let X = X 0 X 1 X m = Y be an MMP as in Theorem 2. K 3 Y 64(b 1(X) + b 3 (X) + b 2 (X)). At each step we want to bound K 3 X i K 3 X i+1 with a topological invariant of X. The number of steps in bounded by b 2. (With flips it is bounded by 2b 2.) Luca Tasin (MPIM) Chern numbers of 3-folds 9 / 19

27 A quick remark Z = P 3 and C a smooth rational curve of degree d. π : W Z be blow-up along C. b 1 (W ) = b 3 (W ) = b 5 (W ) = 0, b 2 (W ) = b 4 (W ) = 2. Luca Tasin (MPIM) Chern numbers of 3-folds 10 / 19

28 A quick remark Z = P 3 and C a smooth rational curve of degree d. π : W Z be blow-up along C. b 1 (W ) = b 3 (W ) = b 5 (W ) = 0, b 2 (W ) = b 4 (W ) = 2. KW 3 = K Z 3 2K Z.C + 2 2g(C) = d The Betti numbers are in general not enough to bound KX 3. Luca Tasin (MPIM) Chern numbers of 3-folds 10 / 19

29 Divisorial contractions to curves I Let f : W Z be a blow-down to a smooth curve with exceptional divisor E. K W = f K Z + E and H 2 (W, Z) = Z[E] H 2 (Z, Z). Luca Tasin (MPIM) Chern numbers of 3-folds 11 / 19

30 Divisorial contractions to curves I Let f : W Z be a blow-down to a smooth curve with exceptional divisor E. K W = f K Z + E and H 2 (W, Z) = Z[E] H 2 (Z, Z). Let E 1,..., E n be the pull-back of a basis of H 2 (Z, Z). E.E i.e j = E i E.E j E = 0 and K 3 W K 3 Z = 2E g(C). Luca Tasin (MPIM) Chern numbers of 3-folds 11 / 19

31 Divisorial contractions to curves I Let f : W Z be a blow-down to a smooth curve with exceptional divisor E. K W = f K Z + E and H 2 (W, Z) = Z[E] H 2 (Z, Z). Let E 1,..., E n be the pull-back of a basis of H 2 (Z, Z). E.E i.e j = E i E.E j E = 0 and K 3 W K 3 Z = 2E g(C). Let x 0,..., x n be coordinates on H 2 (W, Z) with respect to E, E 1,..., E n. Then n F W (x 0,..., x n ) = ax x 0 2 ( b i x i ) + F Z (x 1,..., x n ), where a = E 3 and b i Z. i=1 Luca Tasin (MPIM) Chern numbers of 3-folds 11 / 19

32 An arithmetic result Theorem 3 Let F Z[x 0,..., x n ] be a cubic form such that F 0. Then, modulo the action of GL(Z, n) on (x 1,..., x n ), there are only finitely many triples (a, (b 1,..., b n ), G) such that a, b i Z, G(x 1,..., x n ) is a cubic form and F can be written as Moreover G 0. F = ax ( b i x i )x G(x 1,..., x n ). Luca Tasin (MPIM) Chern numbers of 3-folds 12 / 19

33 An arithmetic result Theorem 3 Let F Z[x 0,..., x n ] be a cubic form such that F 0. Then, modulo the action of GL(Z, n) on (x 1,..., x n ), there are only finitely many triples (a, (b 1,..., b n ), G) such that a, b i Z, G(x 1,..., x n ) is a cubic form and F can be written as Moreover G 0. F = ax ( b i x i )x G(x 1,..., x n ). Define the Skansen number of X as S X := sup{ a : F X may be written in reduced form w.r.t. (a, b, G)}. Luca Tasin (MPIM) Chern numbers of 3-folds 12 / 19

34 Divisorial contractions to curves II Let f : W Z be a blow-down to a smooth curve C of genus g. χ(w \ E) = χ(z \ C) b 3 (W ) = b 3 (Z ) + 2g. Luca Tasin (MPIM) Chern numbers of 3-folds 13 / 19

35 Divisorial contractions to curves II Let f : W Z be a blow-down to a smooth curve C of genus g. χ(w \ E) = χ(z \ C) b 3 (W ) = b 3 (Z ) + 2g. Then KW 3 K Z 3 = 2E g 2S W + 6(b 3 (W ) + 1). Luca Tasin (MPIM) Chern numbers of 3-folds 13 / 19

36 Divisorial contractions to curves II Let f : W Z be a blow-down to a smooth curve C of genus g. χ(w \ E) = χ(z \ C) b 3 (W ) = b 3 (Z ) + 2g. Then K 3 W K 3 Z = 2E g 2S W + 6(b 3 (W ) + 1). The cubic form on Z is determined (up to finitely many possibilities) by the cubic form on W. Luca Tasin (MPIM) Chern numbers of 3-folds 13 / 19

37 Divisorial contractions to points I Let f : W Z be a divisorial contraction to a point in a minimal model program for X. Luca Tasin (MPIM) Chern numbers of 3-folds 14 / 19

38 Divisorial contractions to points I Let f : W Z be a divisorial contraction to a point in a minimal model program for X. K W = f K Z + de, where E is the exceptional divisor and d is a positive rational number. K 3 W K 3 Z = d 3 E 3 Luca Tasin (MPIM) Chern numbers of 3-folds 14 / 19

39 Divisorial contractions to points I Let f : W Z be a divisorial contraction to a point in a minimal model program for X. K W = f K Z + de, where E is the exceptional divisor and d is a positive rational number. K 3 W K 3 Z = d 3 E 3 Using Kawakita s classification, we can prove that where b 2 = b 2 (X). 0 < K 3 W K 3 Z 28 b 2b 2 2, Luca Tasin (MPIM) Chern numbers of 3-folds 14 / 19

40 Divisorial contractions to points II f : W Z divisorial contraction to a point. H 2 (W, Q) = Q[E] H 2 (Z, Q), but H 2 (W, Z)/H 2 (Z, Z) may have torsion. Luca Tasin (MPIM) Chern numbers of 3-folds 15 / 19

41 Divisorial contractions to points II f : W Z divisorial contraction to a point. H 2 (W, Q) = Q[E] H 2 (Z, Q), but H 2 (W, Z)/H 2 (Z, Z) may have torsion. This torsion depends on the singularities of W and Z. Admitting rational coefficients with bounded denominators we have something like F W (x 0,..., x n ) = ax F Z (x 1,..., x n ). Luca Tasin (MPIM) Chern numbers of 3-folds 15 / 19

42 Reduced forms Let F be a cubic form such that F 0. We want to prove that there are only finitely many possible reduced forms for F. Luca Tasin (MPIM) Chern numbers of 3-folds 16 / 19

43 Reduced forms Let F be a cubic form such that F 0. We want to prove that there are only finitely many possible reduced forms for F. 1 By proving that V F = {p P n : rkh F (p) 2} {F 0} is a finite union of points, line, plane conics and plane cubics, we reduce the problem to binary and ternary cubics. Luca Tasin (MPIM) Chern numbers of 3-folds 16 / 19

44 Reduced forms Let F be a cubic form such that F 0. We want to prove that there are only finitely many possible reduced forms for F. 1 By proving that V F = {p P n : rkh F (p) 2} {F 0} is a finite union of points, line, plane conics and plane cubics, we reduce the problem to binary and ternary cubics. 2 To prove our result for binary and ternary cubics we need Siegel and Faltings theorems on the finiteness of integral and rational points on algebraic curves. Luca Tasin (MPIM) Chern numbers of 3-folds 16 / 19

45 A basic case I Let F Z[x, y, z] be a ternary cubic such that 0. The algebra of the invariants of F (under the action of SL(3, Z)) is generated by two polynomials S and T in the coefficients of F and = T 2 64S 3. Luca Tasin (MPIM) Chern numbers of 3-folds 17 / 19

46 A basic case I Let F Z[x, y, z] be a ternary cubic such that 0. The algebra of the invariants of F (under the action of SL(3, Z)) is generated by two polynomials S and T in the coefficients of F and = T 2 64S 3. Assume that S 0 and write F = Ax 3 + (By + Cz)x 2 + G(y, z). Luca Tasin (MPIM) Chern numbers of 3-folds 17 / 19

47 A basic case I Let F Z[x, y, z] be a ternary cubic such that 0. The algebra of the invariants of F (under the action of SL(3, Z)) is generated by two polynomials S and T in the coefficients of F and = T 2 64S 3. Assume that S 0 and write F = Ax 3 + (By + Cz)x 2 + G(y, z). 1 We can assume that G (modulo SL(2, Z)) is fixed. Luca Tasin (MPIM) Chern numbers of 3-folds 17 / 19

48 A basic case I Let F Z[x, y, z] be a ternary cubic such that 0. The algebra of the invariants of F (under the action of SL(3, Z)) is generated by two polynomials S and T in the coefficients of F and = T 2 64S 3. Assume that S 0 and write F = Ax 3 + (By + Cz)x 2 + G(y, z). 1 We can assume that G (modulo SL(2, Z)) is fixed. 2 Passing to a number field K and acting with SL(2, K ) on (y, z) we may assume that G = dy 3 + z 3 and F = Ax 3 + (B 1 y + C 1 z)x 2 + G(y, z). Luca Tasin (MPIM) Chern numbers of 3-folds 17 / 19

49 A basic case I Let F Z[x, y, z] be a ternary cubic such that 0. The algebra of the invariants of F (under the action of SL(3, Z)) is generated by two polynomials S and T in the coefficients of F and = T 2 64S 3. Assume that S 0 and write F = Ax 3 + (By + Cz)x 2 + G(y, z). 1 We can assume that G (modulo SL(2, Z)) is fixed. 2 Passing to a number field K and acting with SL(2, K ) on (y, z) we may assume that G = dy 3 + z 3 and F = Ax 3 + (B 1 y + C 1 z)x 2 + G(y, z). 3 We need to prove that there are only finitely many a, b, c K such that F = ax 3 + (by + cz)x 2 + G. Luca Tasin (MPIM) Chern numbers of 3-folds 17 / 19

50 A basic case II 4 We get S = bcd and T = 27a 2 d 2 + 4b 3 d + 4c 3 d 2. Luca Tasin (MPIM) Chern numbers of 3-folds 18 / 19

51 A basic case II 4 We get S = bcd and T = 27a 2 d 2 + 4b 3 d + 4c 3 d 2. 5 Consider the curve C P 3 given by the ideal I = (Sx 2 3 dx 1x 2, Tx d 2 x 2 0 x 3 4dx 3 1 4d 2 x 3 2 ). Luca Tasin (MPIM) Chern numbers of 3-folds 18 / 19

52 A basic case II 4 We get S = bcd and T = 27a 2 d 2 + 4b 3 d + 4c 3 d 2. 5 Consider the curve C P 3 given by the ideal I = (Sx 2 3 dx 1x 2, Tx d 2 x 2 0 x 3 4dx 3 1 4d 2 x 3 2 ). 6 Since p g (C) = 3, by Faltings theorem we have only a finite number of K -rational points [a, b, c, 1] on C. Luca Tasin (MPIM) Chern numbers of 3-folds 18 / 19

53 Flops Introduction Theorem (Chen-Hacon, 2012) Any MMP of X can be factored into a sequence of divisorial contractions to points (or their inverses), blow-downs to smooth curves and flops. Luca Tasin (MPIM) Chern numbers of 3-folds 19 / 19

54 Flops Introduction Theorem (Chen-Hacon, 2012) Any MMP of X can be factored into a sequence of divisorial contractions to points (or their inverses), blow-downs to smooth curves and flops. Let f : W Z be a flop. W and Z have the same singularities and the same Betti numbers (Kollár). K 3 W = K 3 Z. Luca Tasin (MPIM) Chern numbers of 3-folds 19 / 19

55 Flops Introduction Theorem (Chen-Hacon, 2012) Any MMP of X can be factored into a sequence of divisorial contractions to points (or their inverses), blow-downs to smooth curves and flops. Let f : W Z be a flop. W and Z have the same singularities and the same Betti numbers (Kollár). K 3 W = K 3 Z. What happens to the cubic form? Luca Tasin (MPIM) Chern numbers of 3-folds 19 / 19

ON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE

ON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE ON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE DILETTA MARTINELLI, STEFAN SCHREIEDER, AND LUCA TASIN Abstract. We show that the number of marked minimal models of an n-dimensional

More information

ON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE

ON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE ON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE DILETTA MARTINELLI, STEFAN SCHREIEDER, AND LUCA TASIN Abstract. We show that the number of marked minimal models of an n-dimensional

More information

Finite generation of pluricanonical rings

Finite generation of pluricanonical rings University of Utah January, 2007 Outline of the talk 1 Introduction Outline of the talk 1 Introduction 2 Outline of the talk 1 Introduction 2 3 The main Theorem Curves Surfaces Outline of the talk 1 Introduction

More information

The Cone Theorem. Stefano Filipazzi. February 10, 2016

The Cone Theorem. Stefano Filipazzi. February 10, 2016 The Cone Theorem Stefano Filipazzi February 10, 2016 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will give an overview

More information

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

Geography on 3-folds of General Type. September 9, Meng Chen Fudan University, Shanghai Meng Chen Fudan University, Shanghai September 9, 2010 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

More information

arxiv: v2 [math.ag] 28 Feb 2017

arxiv: v2 [math.ag] 28 Feb 2017 ON THE CHERN NUMBERS OF A SMOOTH THREEFOLD arxiv:1412.1686v2 [math.ag] 28 Feb 2017 PAOLO CASCINI AND LUCA TASIN Abstract. We study the behaviour of Chern numbers of three dimensional terminal varieties

More information

Linear systems and Fano varieties: introduction

Linear systems and Fano varieties: introduction Linear systems and Fano varieties: introduction Caucher Birkar New advances in Fano manifolds, Cambridge, December 2017 References: [B-1] Anti-pluricanonical systems on Fano varieties. [B-2] Singularities

More information

locus such that Γ + F = 0. Then (Y, Γ = Γ + F ) is kawamata log terminal, and K Y + Γ = π (K X + ) + E + F. Pick H ample such that K Y + Γ + H is nef

locus such that Γ + F = 0. Then (Y, Γ = Γ + F ) is kawamata log terminal, and K Y + Γ = π (K X + ) + E + F. Pick H ample such that K Y + Γ + H is nef 17. The MMP Using the cone (16.6) and contraction theorem (16.3), one can define the MMP in all dimensions: (1) Start with a kawamata log terminal pair (X, ). (2) Is K X + nef? Is yes, then stop. (3) Otherwise

More information

Flops connect minimal models

Flops connect minimal models arxiv:0704.03v [math.ag] 8 Apr 2007 Flops connect minimal models Yujiro Kawamata February, 2008 Abstract A result by Birkar-Cascini-Hacon-McKernan together with the boundedness of length of extremal rays

More information

Minimal model program and moduli spaces

Minimal model program and moduli spaces Minimal model program and moduli spaces Caucher Birkar Algebraic and arithmetic structures of moduli spaces, Hokkaido University, Japan 3-7 Sep 2007 Minimal model program In this talk, all the varieties

More information

FACTORING 3-FOLD FLIPS AND DIVISORIAL CONTRACTIONS TO CURVES

FACTORING 3-FOLD FLIPS AND DIVISORIAL CONTRACTIONS TO CURVES FACTORING 3-FOLD FLIPS AND DIVISORIAL CONTRACTIONS TO CURVES JUNGKAI A. CHEN AND CHRISTOPHER D. HACON 1. introduction Flips, flops and divisorial contractions are the elementary birational maps of the

More information

7. Classification of Surfaces The key to the classification of surfaces is the behaviour of the canonical

7. Classification of Surfaces The key to the classification of surfaces is the behaviour of the canonical 7. Classification of Surfaces The key to the classification of surfaces is the behaviour of the canonical divisor. Definition 7.1. We say that a smooth projective surface is minimal if K S is nef. Warning:

More information

ADJUNCTION AND INVERSION OF ADJUNCTION

ADJUNCTION AND INVERSION OF ADJUNCTION ADJUNCTION AND INVERSION OF ADJUNCTION STEFANO FILIPAZZI Abstract. These notes are meant to be a handout for the student seminar in algebraic geometry at the University of Utah. They are a short introduction

More information

BOUNDEDNESS OF MODULI OF VARIETIES OF GENERAL TYPE

BOUNDEDNESS OF MODULI OF VARIETIES OF GENERAL TYPE BOUNDEDNESS OF MODULI OF VARIETIES OF GENERAL TYPE CHRISTOPHER D. HACON, JAMES M C KERNAN, AND CHENYANG XU Abstract. We show that the family of semi log canonical pairs with ample log canonical class and

More information

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

More information

Chern numbers and Hilbert Modular Varieties

Chern numbers and Hilbert Modular Varieties Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point

More information

The geography of irregular surfaces

The geography of irregular surfaces Università di Pisa Classical Algebraic Geometry today M.S.R.I., 1/26 1/30 2008 Summary Surfaces of general type 1 Surfaces of general type 2 3 and irrational pencils Surface = smooth projective complex

More information

Plane quartics and Fano threefolds of genus twelve

Plane quartics and Fano threefolds of genus twelve Plane quartics and Fano threefolds of genus twelve Shigeru MUKAI 288 triangles are strictly biscribed by a plane quartic curve C P 2 = P 2 (C). Two computations of this number will be presented. This number

More information

HOW TO CLASSIFY FANO VARIETIES?

HOW TO CLASSIFY FANO VARIETIES? HOW TO CLASSIFY FANO VARIETIES? OLIVIER DEBARRE Abstract. We review some of the methods used in the classification of Fano varieties and the description of their birational geometry. Mori theory brought

More information

mult V f, where the sum ranges over prime divisor V X. We say that two divisors D 1 and D 2 are linearly equivalent, denoted by sending

mult V f, where the sum ranges over prime divisor V X. We say that two divisors D 1 and D 2 are linearly equivalent, denoted by sending 2. The canonical divisor In this section we will introduce one of the most important invariants in the birational classification of varieties. Definition 2.1. Let X be a normal quasi-projective variety

More information

arxiv: v1 [math.ag] 13 Mar 2019

arxiv: v1 [math.ag] 13 Mar 2019 THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv:1903.05430v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show

More information

Cones of effective two-cycles on toric manifolds

Cones of effective two-cycles on toric manifolds Cones of effective two-cycles on toric manifolds Hiroshi Sato Gifu Shotoku Gakuen University, Japan August 19, 2010 The International Conference GEOMETRY, TOPOLOGY, ALGEBRA and NUMBER THEORY, APPLICATIONS

More information

The structure of algebraic varieties

The structure of algebraic varieties The structure of algebraic varieties János Kollár Princeton University ICM, August, 2014, Seoul with the assistance of Jennifer M. Johnson and Sándor J. Kovács (Written comments added for clarity that

More information

LECTURE 7, WEDNESDAY

LECTURE 7, WEDNESDAY LECTURE 7, WEDNESDAY 25.02.04 FRANZ LEMMERMEYER 1. Singular Weierstrass Curves Consider cubic curves in Weierstraß form (1) E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6, the coefficients a i

More information

AN INFORMAL INTRODUCTION TO THE MINIMAL MODEL PROGRAM

AN INFORMAL INTRODUCTION TO THE MINIMAL MODEL PROGRAM AN INFORMAL INTRODUCTION TO THE MINIMAL MODEL PROGRAM ALEX KÜRONYA This is an expanded version of a talk given in the Fall 2007 Forschungsseminar at the Universität Duisburg-Essen. The purpose of our seminar

More information

PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013

PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013 PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013 1. Problems on moduli spaces The main text for this material is Harris & Morrison Moduli of curves. (There are djvu files

More information

MORI DREAM SPACES. Contents. 1. Hilbert s 14th Problem

MORI DREAM SPACES. Contents. 1. Hilbert s 14th Problem MORI DREAM SPACES JAMES M C KERNAN Abstract. We explore the circle of ideas connecting finite generation of the Cox ring, Mori dream spaces and invariant theory. Contents 1. Hilbert s 14th Problem 1 1.1.

More information

LECTURES ON SINGULARITIES AND ADJOINT LINEAR SYSTEMS

LECTURES ON SINGULARITIES AND ADJOINT LINEAR SYSTEMS LECTURES ON SINGULARITIES AND ADJOINT LINEAR SYSTEMS LAWRENCE EIN Abstract. 1. Singularities of Surfaces Let (X, o) be an isolated normal surfaces singularity. The basic philosophy is to replace the singularity

More information

Riemann Surfaces and Algebraic Curves

Riemann Surfaces and Algebraic Curves Riemann Surfaces and Algebraic Curves JWR Tuesday December 11, 2001, 9:03 AM We describe the relation between algebraic curves and Riemann surfaces. An elementary reference for this material is [1]. 1

More information

STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES

STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES DAWEI CHEN AND IZZET COSKUN Contents 1. Introduction 1 2. Preliminary definitions and background 3 3. Degree two maps to Grassmannians 4 4.

More information

Each is equal to CP 1 minus one point, which is the origin of the other: (C =) U 1 = CP 1 the line λ (1, 0) U 0

Each is equal to CP 1 minus one point, which is the origin of the other: (C =) U 1 = CP 1 the line λ (1, 0) U 0 Algebraic Curves/Fall 2015 Aaron Bertram 1. Introduction. What is a complex curve? (Geometry) It s a Riemann surface, that is, a compact oriented twodimensional real manifold Σ with a complex structure.

More information

Introduction to Arithmetic Geometry

Introduction to Arithmetic Geometry Introduction to Arithmetic Geometry 18.782 Andrew V. Sutherland September 5, 2013 What is arithmetic geometry? Arithmetic geometry applies the techniques of algebraic geometry to problems in number theory

More information

Birational geometry of algebraic varieties

Birational geometry of algebraic varieties May-June, 2018 Outline of the talk Singularities of the MMP The Cone Theorem Running the MMP Goals of this lecture series 1 Introduction Singularities of the MMP The Cone Theorem Running the MMP Goals

More information

arxiv:math/ v1 [math.ag] 29 Nov 2003

arxiv:math/ v1 [math.ag] 29 Nov 2003 arxiv:math/0312012v1 [math.ag] 29 Nov 2003 A REMARK ON FANO THREEFOLDS WITH CANONICAL GORENSTEIN SINGULARITIES YURI G. PROKHOROV Abstract. We give some rationality constructions for Fano threefolds with

More information

Quadratic families of elliptic curves and degree 1 conic bundles

Quadratic families of elliptic curves and degree 1 conic bundles Quadratic families of elliptic curves and degree 1 conic bundles János Kollár Princeton University joint with Massimiliano Mella Elliptic curves E := ( y 2 = a 3 x 3 + a 2 x 2 + a 1 x + a 0 ) A 2 xy Major

More information

COMPLEX ALGEBRAIC SURFACES CLASS 9

COMPLEX ALGEBRAIC SURFACES CLASS 9 COMPLEX ALGEBRAIC SURFACES CLASS 9 RAVI VAKIL CONTENTS 1. Construction of Castelnuovo s contraction map 1 2. Ruled surfaces 3 (At the end of last lecture I discussed the Weak Factorization Theorem, Resolution

More information

Basic Algebraic Geometry 1

Basic Algebraic Geometry 1 Igor R. Shafarevich Basic Algebraic Geometry 1 Second, Revised and Expanded Edition Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest Table of Contents Volume 1

More information

Singularities of (L)MMP and applications. Massimiliano Mella. Dipartimento di Matematica, Università di Ferrara, Italia address:

Singularities of (L)MMP and applications. Massimiliano Mella. Dipartimento di Matematica, Università di Ferrara, Italia  address: Singularities of (L)MMP and applications Massimiliano Mella Dipartimento di Matematica, Università di Ferrara, Italia E-mail address: mll@unife.it 1991 Mathematics Subject Classification. Primary 14E30;

More information

Recall for an n n matrix A = (a ij ), its trace is defined by. a jj. It has properties: In particular, if B is non-singular n n matrix,

Recall for an n n matrix A = (a ij ), its trace is defined by. a jj. It has properties: In particular, if B is non-singular n n matrix, Chern characters Recall for an n n matrix A = (a ij ), its trace is defined by tr(a) = n a jj. j=1 It has properties: tr(a + B) = tr(a) + tr(b), tr(ab) = tr(ba). In particular, if B is non-singular n n

More information

Notes for Curves and Surfaces Instructor: Robert Freidman

Notes for Curves and Surfaces Instructor: Robert Freidman Notes for Curves and Surfaces Instructor: Robert Freidman Henry Liu April 25, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR8293 Algebraic Curves & Surfaces. Let me

More information

ON SHOKUROV S RATIONAL CONNECTEDNESS CONJECTURE

ON SHOKUROV S RATIONAL CONNECTEDNESS CONJECTURE ON SHOKUROV S RATIONAL CONNECTEDNESS CONJECTURE CHRISTOPHER D. HACON AND JAMES M C KERNAN Abstract. We prove a conjecture of V. V. Shokurov which in particular implies that the fibers of a resolution of

More information

AN INTRODUCTION TO ARITHMETIC AND RIEMANN SURFACE. We describe points on the unit circle with coordinate satisfying

AN INTRODUCTION TO ARITHMETIC AND RIEMANN SURFACE. We describe points on the unit circle with coordinate satisfying AN INTRODUCTION TO ARITHMETIC AND RIEMANN SURFACE 1. RATIONAL POINTS ON CIRCLE We start by asking us: How many integers x, y, z) can satisfy x 2 + y 2 = z 2? Can we describe all of them? First we can divide

More information

ON SINGULAR CUBIC SURFACES

ON SINGULAR CUBIC SURFACES ASIAN J. MATH. c 2009 International Press Vol. 13, No. 2, pp. 191 214, June 2009 003 ON SINGULAR CUBIC SURFACES IVAN CHELTSOV Abstract. We study global log canonical thresholds of singular cubic surfaces.

More information

FAKE PROJECTIVE SPACES AND FAKE TORI

FAKE PROJECTIVE SPACES AND FAKE TORI FAKE PROJECTIVE SPACES AND FAKE TORI OLIVIER DEBARRE Abstract. Hirzebruch and Kodaira proved in 1957 that when n is odd, any compact Kähler manifold X which is homeomorphic to P n is isomorphic to P n.

More information

HODGE NUMBERS OF COMPLETE INTERSECTIONS

HODGE NUMBERS OF COMPLETE INTERSECTIONS HODGE NUMBERS OF COMPLETE INTERSECTIONS LIVIU I. NICOLAESCU 1. Holomorphic Euler characteristics Suppose X is a compact Kähler manifold of dimension n and E is a holomorphic vector bundle. For every p

More information

arxiv:math/ v1 [math.ag] 6 Aug 2002

arxiv:math/ v1 [math.ag] 6 Aug 2002 arxiv:math/0208049v1 [math.ag] 6 Aug 2002 1 Introduction Semistable divisorial contractions Paul Hacking August 5th, 2002 The semistable minimal model program is a special case of the minimal model program

More information

The sectional genus of quasi-polarised varieties

The sectional genus of quasi-polarised varieties The sectional genus of quasi-polarised varieties Andreas Höring Abstract. T. Fujita conjectured that the sectional genus of a quasi-polarised variety is non-negative. We prove this conjecture. Using the

More information

Special cubic fourfolds

Special cubic fourfolds Special cubic fourfolds 1 Hodge diamonds Let X be a cubic fourfold, h H 2 (X, Z) be the (Poincaré dual to the) hyperplane class. We have h 4 = deg(x) = 3. By the Lefschetz hyperplane theorem, one knows

More information

Recent Developments in the Classification Theory of Compact Kähler Manifolds

Recent Developments in the Classification Theory of Compact Kähler Manifolds Several Complex Variables MSRI Publications Volume 37, 1999 Recent Developments in the Classification Theory of Compact Kähler Manifolds FRÉDÉRIC CAMPANA AND THOMAS PETERNELL Abstract. We review some of

More information

Moduli spaces of log del Pezzo pairs and K-stability

Moduli spaces of log del Pezzo pairs and K-stability Report on Research in Groups Moduli spaces of log del Pezzo pairs and K-stability June 20 - July 20, 2016 Organizers: Patricio Gallardo, Jesus Martinez-Garcia, Cristiano Spotti. In this report we start

More information

arxiv:math/ v3 [math.ag] 1 Mar 2006

arxiv:math/ v3 [math.ag] 1 Mar 2006 arxiv:math/0506132v3 [math.ag] 1 Mar 2006 A NOTE ON THE PROJECTIVE VARIETIES OF ALMOST GENERAL TYPE SHIGETAKA FUKUDA Abstract. A Q-Cartier divisor D on a projective variety M is almost nup, if (D, C) >

More information

7.3 Singular Homology Groups

7.3 Singular Homology Groups 184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the

More information

Riemann surfaces. Paul Hacking and Giancarlo Urzua 1/28/10

Riemann surfaces. Paul Hacking and Giancarlo Urzua 1/28/10 Riemann surfaces Paul Hacking and Giancarlo Urzua 1/28/10 A Riemann surface (or smooth complex curve) is a complex manifold of dimension one. We will restrict to compact Riemann surfaces. It is a theorem

More information

L6: Almost complex structures

L6: Almost complex structures L6: Almost complex structures To study general symplectic manifolds, rather than Kähler manifolds, it is helpful to extract the homotopy-theoretic essence of having a complex structure. An almost complex

More information

LOG MINIMAL MODELS ACCORDING TO SHOKUROV

LOG MINIMAL MODELS ACCORDING TO SHOKUROV LOG MINIMAL MODELS ACCORDING TO SHOKUROV CAUCHER BIRKAR Abstract. Following Shokurov s ideas, we give a short proof of the following klt version of his result: termination of terminal log flips in dimension

More information

Smoothable del Pezzo surfaces with quotient singularities

Smoothable del Pezzo surfaces with quotient singularities Smoothable del Pezzo surfaces with quotient singularities Paul Hacking and Yuri Prokhorov Abstract We classify del Pezzo surfaces with quotient singularities and Picard rank which admit a Q-Gorenstein

More information

Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds

Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds Yukinobu Toda Kavli IPMU, University of Tokyo Yukinobu Toda (Kavli IPMU) Birational geometry 1 / 38 1. Motivation Yukinobu

More information

EXISTENCE OF MINIMAL MODELS FOR VARIETIES OF LOG GENERAL TYPE

EXISTENCE OF MINIMAL MODELS FOR VARIETIES OF LOG GENERAL TYPE EXISTENCE OF MINIMAL MODELS FOR VARIETIES OF LOG GENERAL TYPE CAUCHER BIRKAR, PAOLO CASCINI, CHRISTOPHER D. HACON, AND JAMES M C KERNAN Abstract. We prove that the canonical ring of a smooth projective

More information

ACC FOR LOG CANONICAL THRESHOLDS

ACC FOR LOG CANONICAL THRESHOLDS ACC FOR LOG CANONICAL THRESHOLDS CHRISTOPHER D. HACON, JAMES M C KERNAN, AND CHENYANG XU To Vyacheslav Shokurov on the occasion of his sixtieth birthday Abstract. We show that log canonical thresholds

More information

CAUCHER BIRKAR. Abstract. We prove that the Iitaka conjecture C n,m for algebraic fibre spaces holds up to dimension 6, that is, when n 6.

CAUCHER BIRKAR. Abstract. We prove that the Iitaka conjecture C n,m for algebraic fibre spaces holds up to dimension 6, that is, when n 6. IITAKA CONJECTURE C n,m IN DIMENSION SIX CAUCHER BIRKAR Abstract. We prove that the Iitaka conjecture C n,m for algebraic fibre spaces holds up to dimension 6, that is, when n 6. 1. Introduction We work

More information

Diagonal Subschemes and Vector Bundles

Diagonal Subschemes and Vector Bundles Pure and Applied Mathematics Quarterly Volume 4, Number 4 (Special Issue: In honor of Jean-Pierre Serre, Part 1 of 2 ) 1233 1278, 2008 Diagonal Subschemes and Vector Bundles Piotr Pragacz, Vasudevan Srinivas

More information

arxiv:math/ v1 [math.ag] 11 Apr 2003 James McKernan and Yuri Prokhorov

arxiv:math/ v1 [math.ag] 11 Apr 2003 James McKernan and Yuri Prokhorov THREEFOLD THRESHOLDS arxiv:math/0304152v1 [math.ag] 11 Apr 2003 James McKernan and Yuri Prokhorov Abstract. We prove that the set of accumulation points of thresholds in dimension three is equal to the

More information

arxiv: v4 [math.ag] 16 Mar 2019

arxiv: v4 [math.ag] 16 Mar 2019 A NOTE ON A SMOOTH PROJECTIVE SURFACE WITH PICARD NUMBER 2 SICHEN LI Abstract. We characterize the integral Zariski decomposition of a smooth projective surface with Picard number 2 to partially solve

More information

Some Remarks on the Minimal Model Program

Some Remarks on the Minimal Model Program J. Math. Sci. Univ. Tokyo 22 (2015), 149 192. Some Remarks on the Minimal Model Program for Log Canonical Pairs By Osamu Fujino In memory of Professor Kunihiko Kodaira Abstract. We prove that the target

More information

Linear systems and Fano varieties: BAB, and related topics

Linear systems and Fano varieties: BAB, and related topics Linear systems and Fano varieties: BAB, and related topics Caucher Birkar New advances in Fano manifolds, Cambridge, December 2017 References: [B-1] Anti-pluricanonical systems on Fano varieties. [B-2]

More information

Introduction Curves Surfaces Curves on surfaces. Curves and surfaces. Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway

Introduction Curves Surfaces Curves on surfaces. Curves and surfaces. Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway Curves and surfaces Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway What is algebraic geometry? IMA, April 13, 2007 Outline Introduction Curves Surfaces Curves on surfaces

More information

Holomorphic symmetric differentials and a birational characterization of Abelian Varieties

Holomorphic symmetric differentials and a birational characterization of Abelian Varieties Holomorphic symmetric differentials and a birational characterization of Abelian Varieties Ernesto C. Mistretta Abstract A generically generated vector bundle on a smooth projective variety yields a rational

More information

Diophantine equations and beyond

Diophantine equations and beyond Diophantine equations and beyond lecture King Faisal prize 2014 Gerd Faltings Max Planck Institute for Mathematics 31.3.2014 G. Faltings (MPIM) Diophantine equations and beyond 31.3.2014 1 / 23 Introduction

More information

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Application of cohomology: Hilbert polynomials and functions, Riemann- Roch, degrees, and arithmetic genus 1 1. APPLICATION OF COHOMOLOGY:

More information

COMPLEX ALGEBRAIC SURFACES CLASS 6

COMPLEX ALGEBRAIC SURFACES CLASS 6 COMPLEX ALGEBRAIC SURFACES CLASS 6 RAVI VAKIL CONTENTS 1. The intersection form 1.1. The Neron-Severi group 3 1.. Aside: The Hodge diamond of a complex projective surface 3. Riemann-Roch for surfaces 4

More information

An invitation to log geometry p.1

An invitation to log geometry p.1 An invitation to log geometry James M c Kernan UCSB An invitation to log geometry p.1 An easy integral Everyone knows how to evaluate the following integral: 1 0 1 1 x 2 dx. An invitation to log geometry

More information

EXISTENCE OF MINIMAL MODELS FOR VARIETIES OF LOG GENERAL TYPE

EXISTENCE OF MINIMAL MODELS FOR VARIETIES OF LOG GENERAL TYPE EXISTENCE OF MINIMAL MODELS FOR VARIETIES OF LOG GENERAL TYPE CAUCHER BIRKAR, PAOLO CASCINI, CHRISTOPHER D. HACON, AND JAMES M C KERNAN Abstract. We prove that the canonical ring of a smooth projective

More information

On a connectedness principle of Shokurov-Kollár type

On a connectedness principle of Shokurov-Kollár type . ARTICLES. SCIENCE CHINA Mathematics March 2019 Vol. 62 No. 3: 411 416 https://doi.org/10.1007/s11425-018-9360-5 On a connectedness principle of Shokurov-Kollár type Christopher D. Hacon 1 & Jingjun Han

More information

arxiv: v1 [math.ag] 3 Mar 2009

arxiv: v1 [math.ag] 3 Mar 2009 TOWARDS BOUNDEDNESS OF MINIMAL LOG DISCREPANCIES BY RIEMANN ROCH THEOREM MASAYUKI KAWAKITA arxiv:0903.0418v1 [math.ag] 3 Mar 2009 ABSTRACT. We introduce an approach of Riemann Roch theorem to the boundedness

More information

Bjorn Poonen. MSRI Introductory Workshop on Rational and Integral Points on Higher-dimensional Varieties. January 17, 2006

Bjorn Poonen. MSRI Introductory Workshop on Rational and Integral Points on Higher-dimensional Varieties. January 17, 2006 University of California at Berkeley MSRI Introductory Workshop on Rational and Integral Points on Higher-dimensional (organized by Jean-Louis Colliot-Thélène, Roger Heath-Brown, János Kollár,, Alice Silverberg,

More information

Computing Fano 3-folds of index 3

Computing Fano 3-folds of index 3 arxiv:math/0610958v1 [math.ag] 31 Oct 2006 Computing Fano 3-folds of index 3 Gavin Brown Abstract Kaori Suzuki We use the computer algebra system Magma to study graded rings of Fano 3-folds of index 3

More information

K-EQUIVALENCE IN BIRATIONAL GEOMETRY

K-EQUIVALENCE IN BIRATIONAL GEOMETRY K-EQUIVALENCE IN BIRATIONAL GEOMETRY CHIN-LUNG WANG In this article we survey the background and recent development on the K- equivalence relation among birational manifolds. The content is based on the

More information

Then the blow up of V along a line is a rational conic bundle over P 2. Definition A k-rational point of a scheme X over S is any point which

Then the blow up of V along a line is a rational conic bundle over P 2. Definition A k-rational point of a scheme X over S is any point which 16. Cubics II It turns out that the question of which varieties are rational is one of the subtlest geometric problems one can ask. Since the problem of determining whether a variety is rational or not

More information

RESEARCH STATEMENT FEI XU

RESEARCH STATEMENT FEI XU RESEARCH STATEMENT FEI XU. Introduction Recall that a variety is said to be rational if it is birational to P n. For many years, mathematicians have worked on the rationality of smooth complete intersections,

More information

where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism

where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism 8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the

More information

Complex Algebraic Geometry: Smooth Curves Aaron Bertram, First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool

Complex Algebraic Geometry: Smooth Curves Aaron Bertram, First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool Complex Algebraic Geometry: Smooth Curves Aaron Bertram, 2010 12. First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool for classifying smooth projective curves, i.e. giving

More information

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1)

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1) Tuesday 10 February 2004 (Day 1) 1a. Prove the following theorem of Banach and Saks: Theorem. Given in L 2 a sequence {f n } which weakly converges to 0, we can select a subsequence {f nk } such that the

More information

arxiv: v1 [math.ag] 10 Jun 2016

arxiv: v1 [math.ag] 10 Jun 2016 THE EVENTUAL PARACANONICAL MAP OF A VARIETY OF MAXIMAL ALBANESE DIMENSION arxiv:1606.03301v1 [math.ag] 10 Jun 2016 MIGUEL ÁNGEL BARJA, RITA PARDINI AND LIDIA STOPPINO Abstract. Let X be a smooth complex

More information

STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES

STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES DAWEI CHEN AND IZZET COSKUN Abstract. In this paper, we determine the stable base locus decomposition of the Kontsevich moduli spaces of degree

More information

LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY

LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in

More information

Threefold Flops and the Contraction Algebra

Threefold Flops and the Contraction Algebra Threefold Flops and the Contraction Algebra Matt Booth March 2017 April 2018: added some references, corrected a typo, made some minor rearrangements. 1 Basic birational geometry I ll work with irreducible

More information

IMAGES OF MANIFOLDS WITH SEMI-AMPLE ANTI-CANONICAL DIVISOR

IMAGES OF MANIFOLDS WITH SEMI-AMPLE ANTI-CANONICAL DIVISOR IMAGES OF MANIFOLDS WITH SEMI-AMPLE ANTI-CANONICAL DIVISOR CAUCHER BIRKAR YIFEI CHEN Abstract. We prove that if f : X Z is a smooth surjective morphism between projective manifolds and if K X is semi-ample,

More information

arxiv: v1 [math.ag] 2 Apr 2015

arxiv: v1 [math.ag] 2 Apr 2015 The Sarkisov program for Mori fibred Calabi Yau pairs ariv:1504.00557v1 [math.ag] 2 Apr 2015 Alessio Corti Department of Mathematics Imperial College London 180 Queen s Gate London, SW7 2AZ, UK July 30,

More information

The Theorem of Gauß-Bonnet in Complex Analysis 1

The Theorem of Gauß-Bonnet in Complex Analysis 1 The Theorem of Gauß-Bonnet in Complex Analysis 1 Otto Forster Abstract. The theorem of Gauß-Bonnet is interpreted within the framework of Complex Analysis of one and several variables. Geodesic triangles

More information

Some examples of Calabi-Yau pairs with maximal intersection and no toric model

Some examples of Calabi-Yau pairs with maximal intersection and no toric model Some examples of Calabi-Yau pairs with maximal intersection and no toric model Anne-Sophie Kaloghiros Abstract It is known that a maximal intersection log canonical Calabi-Yau surface pair is crepant birational

More information

Under these assumptions show that if D X is a proper irreducible reduced curve with C K X = 0 then C is a smooth rational curve.

Under these assumptions show that if D X is a proper irreducible reduced curve with C K X = 0 then C is a smooth rational curve. Example sheet 3. Positivity in Algebraic Geometry, L18 Instructions: This is the third and last example sheet. More exercises may be added during this week. The final example class will be held on Thursday

More information

Singularities of linear systems and boundedness of Fano varieties. Caucher Birkar

Singularities of linear systems and boundedness of Fano varieties. Caucher Birkar Singularities of linear systems and boundedness of Fano varieties Caucher Birkar Abstract. We study log canonical thresholds (also called global log canonical threshold or α-invariant) of R-linear systems.

More information

The Riemann-Roch Theorem

The Riemann-Roch Theorem The Riemann-Roch Theorem Paul Baum Penn State TIFR Mumbai, India 20 February, 2013 THE RIEMANN-ROCH THEOREM Topics in this talk : 1. Classical Riemann-Roch 2. Hirzebruch-Riemann-Roch (HRR) 3. Grothendieck-Riemann-Roch

More information

THE VIRTUAL SINGULARITY THEOREM AND THE LOGARITHMIC BIGENUS THEOREM. (Received October 28, 1978, revised October 6, 1979)

THE VIRTUAL SINGULARITY THEOREM AND THE LOGARITHMIC BIGENUS THEOREM. (Received October 28, 1978, revised October 6, 1979) Tohoku Math. Journ. 32 (1980), 337-351. THE VIRTUAL SINGULARITY THEOREM AND THE LOGARITHMIC BIGENUS THEOREM SHIGERU IITAKA (Received October 28, 1978, revised October 6, 1979) Introduction. When we study

More information

FANO THREEFOLDS WITH LARGE AUTOMORPHISM GROUPS

FANO THREEFOLDS WITH LARGE AUTOMORPHISM GROUPS FANO THREEFOLDS WITH LARGE AUTOMORPHISM GROUPS CONSTANTIN SHRAMOV Let G be a finite group and k a field. We can consider the notion of G-rationality, G- nonrationality, etc., by considering G-equivariant

More information

Oral exam practice problems: Algebraic Geometry

Oral exam practice problems: Algebraic Geometry Oral exam practice problems: Algebraic Geometry Alberto García Raboso TP1. Let Q 1 and Q 2 be the quadric hypersurfaces in P n given by the equations f 1 x 2 0 + + x 2 n = 0 f 2 a 0 x 2 0 + + a n x 2 n

More information

arxiv: v2 [math.dg] 6 Nov 2014

arxiv: v2 [math.dg] 6 Nov 2014 A SHARP CUSP COUNT FOR COMPLEX HYPERBOLIC SURFACES AND RELATED RESULTS GABRIELE DI CERBO AND LUCA F. DI CERBO arxiv:1312.5368v2 [math.dg] 6 Nov 2014 Abstract. We derive a sharp cusp count for finite volume

More information

Vanishing theorems and holomorphic forms

Vanishing theorems and holomorphic forms Vanishing theorems and holomorphic forms Mihnea Popa Northwestern AMS Meeting, Lansing March 14, 2015 Holomorphic one-forms and geometry X compact complex manifold, dim C X = n. Holomorphic one-forms and

More information

Rigidity properties of Fano varieties

Rigidity properties of Fano varieties Current Developments in Algebraic Geometry MSRI Publications Volume 59, 2011 Rigidity properties of Fano varieties TOMMASO DE FERNEX AND CHRISTOPHER D. HACON We overview some recent results on Fano varieties

More information

On the Virtual Fundamental Class

On the Virtual Fundamental Class On the Virtual Fundamental Class Kai Behrend The University of British Columbia Seoul, August 14, 2014 http://www.math.ubc.ca/~behrend/talks/seoul14.pdf Overview Donaldson-Thomas theory: counting invariants

More information