Cubic fourfolds and odd-torsion Brauer Manin obstructions on K3 surfaces
|
|
- Alexandrina Kristina Hensley
- 5 years ago
- Views:
Transcription
1 Cubic fourfolds and odd-torsion Brauer Manin obstructions on K3 surfaces Anthony Várilly-Alvarado Rice University Arithmetic Geometry, Number Theory and Computation August 22nd, 2018
2 We report on results of a collaboration Everything today is joint work with Jen Berg.
3 Brauer Manin recap X /k a nice variety over a number field. H Br(X ) := H 2 et(x,g m ) tors gives rise to an obstruction set X (k) X (A) H X (A) := v X (k v ) that contains X (k) (for product topology). Hasse principle: X (A) }{{} = X (k) }{{} locally soluble globally soluble Weak Approximation: locally soluble + X (k) = X (A).
4 Brauer Manin recap Brauer Manin obstruction to Hasse Principle: X (A) but X (A) H = for some H Br(X ). Brauer Manin obstruction to Weak Approximation: X (A) \ X (A) H for some H Br(X ).
5 K3 surfaces K3 surface: nice surface X with ω X O X and H 1 (X,O X ) = 0. Example A smooth double cover X P 2, branched along a plane sextic: w 2 = f 6 (x,y,z) This is a degree 2 K3 surface. Conjecture (Skorobogatov, 2009) If X /k is a locally soluble K3 surface over a number field, then X (k) = X (A) Br(X ). Informally: the Brauer Manin obstruction accounts for failures of the Hasse Principle and Weak Approximation on K3 surfaces.
6 A brief history of K3 obstructions X /k a K3 surface over a number field. H = Br(X )[2] can obstruct Weak Approximation: Wittenberg, 2004 Ieronymou, Hassett, V.-A., Elsenhans, Jahnel, Mckinnie, Sawon, Tanimoto, V.-A., H = Br(X )[2] can obstruct the Hasse principle: Martin Bright, Hassett, V.-A., H = Br(X )[3] or Br(X )[5] can obstruct the Weak Approximation: Preu, Ieronymou Skorobogatov, 2015.
7 What about odd torsion and the Hasse Principle? Question (Ieronymou Skorobogatov, 2015) Is there a locally soluble K3 surface X over a number field with X (A) Br(X ) odd =? Theorem (Corn Nakahara, 2017) The degree 2 K3 surface X /Q w 2 = 3x 6 +97y z 6 is locally soluble. The cyclic algebra ( A := Q( 3 28,ζ 3 )/Q(ζ 3 ), w ) 3x 3 w + BrQ(ζ 3 )(X ) 3x 3 extends to an element of Br(X Q(ζ3 )) that gives rise to a Brauer-Manin obstruction to the Hasse Principle on X.
8 What about odd torsion in the Brauer group? The class A in Corn Nakahara is algebraic. Recall there is a filtration Br 0 (X ) Br 1 (X ) Br(X ). }{{}}{{} im(brk BrX ) ker(br(x ) Br(X )) constant classes algebraic classes = X (A) Br 0(X ) = X (A), so we often consider the quotients Br 1 (X )/Br 0 (X ) and Br(X )/Br 1 (X ) when computing Brauer Manin obstructions.
9 Can odd transcendental classes obstruct HP on a K3? Theorem (Berg, V.-A., 2018) There is a degree 2 K3 surface X /Q, together with an A Br(X )[3], such that X (A) and X (A) A =. There is no algebraic Brauer Manin obstruction to the Hasse Principle on X. For last claim: Pic(X ) Z, and hence Br 1 (X )/Br 0 (X ) = 0.
10 Can odd transcendental classes obstruct HP on a K3? How can we find transcendental 3-torsion in Br(X )? X /C: a complex projective K3 surface with NS(X ) = Zh, h 2 = 2d. Let T X := NS(X ) H 2 (X,Z) = U 3 E 8 ( 1) 2 =: Λ K3 be the transcendental lattice of X. Br(X ) TX Q/Z, so there is a one-to-one correspondence { A BrX of order n} 1 1 {surjections T X Z/nZ} Hence, to A as above, we may associate T A T X : T A = ker(a : T X Z/nZ).
11 Sublattices of index 3 in T X Theorem (Mckinnie, Sawon, Tanimoto, V.-A., 2017) Let X be a complex projective K3 surface with PicX = Zh, h 2 = 2, and let A (BrX )[3]. Then either 1. there is a unique primitive embedding T A Λ K3. This gives a degree 18 K3 surface Y associated to the pair (X, A ); OR 2. T A ( 1) = H 2,T H 4 (Y,Z), where Y is a cubic fourfold of discriminant 18 (H is the hyperplane class); OR, 3. T A ( 1) is a lattice with discriminant group Z/2Z (Z/3Z) 2.
12 Cubic fourfolds of discriminant 18 Hence, lattice theory suggests: Y cubic fourfold of discriminant 18 (X, A ), where X is a K3 surface of degree 2 and A Br(X )[3]. A cubic fourfold of discriminant 18 is a smooth cubic fourfold Y P 5, together with a rank two saturated lattice H 2,T H 2,2 (Y ) H 4 (Y,Z) of discriminant 18, where H is the hyperplane class, T H T 6 18 H 2 Hassett, 2000: such fourfolds exist.
13 What is the surface T? Theorem (Addington, Hassett, Tschinkel, V.-A., 2016) A general cubic fourfold Y of discriminant 18 contains an elliptic ruled surface T of degree 6. The linear system of quadrics in P 5 containing T is 3-dimensional. Let Q 0,Q 1,Q 2 C[x 0,...,x 5 ] 2 be a basis for H 0 (T,I T (2)) Bl T (Y ) π Y P(H 0 (T,I T (2)) ) = P 2 x = [x 0,...,x 5 ] [Q 0 ( x),q 1 ( x),q 2 ( x)]
14 Fibers of π Key insight: General fiber of π: Bl T (Y ) P 2 is a del Pezzo surface of degree 6. Geometrically, a dp6 is a blow-up of P 2 at 3 non-colinear points. The locus of fibers that are geometrically blow-ups of P 2 at three distinct colinear points has image a sextic curve under π: C : f (x,y,z) = 0. The extension K := C(P 2 )( f ) splits the two triples of pairwise skew exceptional curves in the generic fiber S of π. K is the function field of our K3 surface X of degree 2!
15 Brauer elements of order 3 Let S K be the (dp6) generic fiber of π base extended to K. S K contains two triples of pairwise skew exceptional curves: {E 1,E 2,E 3 } and {L E 1 E 2,L E 2 E 3,L E 1 E 3 }. These can be blown-down, respectively, with the morphisms associated to the linear systems 3L and 3(2L E 1 E 2 E 3 ).
16 Brauer elements of order 3 We get φ 3L : S K X 1 and φ 3(2L E1 E 2 E 3 ) : S K X 2 X 1 and X 2 are Severi-Brauer varieties of dimension 2 over K. Hence X 1 and X 2 give rise to two classes A 1 and A 2 (BrK)[3]. Proposition (Corn, 2005) A 1 A 2 = Id in BrK. Combining work of Corn, Kollár, AHTVA, and Kuznetsov, we can spread these classes to our K3 surface X. We have recovered a pair (X, { Id,A 1,A 2 } )!
17 Brauer elements of order 3: alternatively... In each smooth fiber S of π, there are two families of twisted cubics, each one two dimensional, parametrized by P(H 0 (S,O S (L))) = P 2 and P(H 0 (S,O S (2L E 1 E 2 E 3 ))) = P 2. These two P 2 s come together over the locus f (x,y,z) = 0 of the base P 2 of π. Let H be the relative Hilbert scheme that parametrizes twisted cubics in the fibers of π. The Stein factorization of H P 2 is H X P 2 where X is our K3 surface, and H is an étale P 2 bundle over X.
18 Brauer elements of order 3: alternatively... Hence H gives rise to an element A Br(X [3]). Notes: 1. The covering involution ι: X X sending w w induces ι : Br(X ) Br(X ) sending A A 1, so H actually encodes the group A. 2. If (Y,T ) are defined over a field k (e.g., a number field), then so are H and X.
19 Computing the obstruction Theorem (Wedderburn) Every central simple algebra of degree 3 over a field is cyclic. In principle, can write down cyclic representatives for A 1 and A 2. Challenge Do it.
20 Computing the obstruction { X (A) A = (P v ) X (A) : v } inv v A (P v ) = 0. inv v A (P v ) = 0 the fiber in H above P v X (k v ) is a split P 2 (i.e., is isomorphic to P 2 over k v ) the fiber has a k v -rational point. From now on, X is defined over Q. To produce an example with X (A) A =, we rig (X,A ) so that inv v A (P v ) = { 0 if v 3, 1/3 or 2/3 if v = 3.
21 Lang Nishimura to the rescue Lemma (Lang Nishimura) Let Z 1 and Z 2 be birational nice Q v -varieties. Then Z 1 (Q v ) Z 2 (Q v ). The fiber in H above P v X (Q v ) is birational to the dp6 fiber of π above the image of P v in P 2. Apply Lang-Nishimura: to have inv 3 A (P) 0 for all P X (Q 3 ), it suffices to have Bl T (Y )(Q 3 ) =. Applying Lang-Nishimura again, it suffices to have Y (Q 3 ) =.
22 Construction of the cubic fourfold Thus, the hardest part of our task is to produce a cubic fourfold Y /Q of discriminant 18, such that Y (Q 3 ) =. Let P 5 := ProjQ[x 0,x 1,x 2,x 3,x 4,x 5 ], and define quadrics cut out by Q 1 := x 0 x 3 +x 2 x 3 x 0 x 4 +x 1 x 4 +3x 2 x 4 +5x 0 x 5 x 1 x 5 Q 2 := x 1 x 3 +5x 0 x 4 2x 2 x 4 2x 0 x 5 +5x 1 x 5 +x 2 x 5 Q 3 := 2x 2 x 3 x 0 x 4 2x 1 x 4 2x 2 x 4 +x 1 x 5 Each quadric contains the planes Π 1 = {x 0 = x 1 = x 2 = 0} and Π 2 = {x 3 = x 4 = x 5 = 0}.
23 Construction of the cubic fourfold The sextic elliptic surface T is obtained by saturating the ideal Q 1,Q 2,Q 3 with respect to I (Π 1 )I (Π 2 ). It is cut out by Q 1,Q 2,Q 3 and the two cubics C 1 := 2x3 3 +5x 3 2 x 4 +x 3 x x x 3 2 x 5 26x 3 x 4 x 5 11x4 2 x 5 +47x 3 x x 4x5 2 +5x 5 3, C 2 := 2x0 3 x 0 2 x 1 2x 0 x1 2 x x 0 2 x 2 +10x 0 x 1 x 2 +x1 2 x 2 11x 0 x2 2 18x 1x2 2 4x 2 3 Magical property: The plane cubics, {C 1 = 0} and {C 2 = 0} have no Z/3Z-points! Bhargava, Cremona, Fisher: 99.25% of plane cubics have Q 3 -points.
24 Construction of the cubic fourfold C 1 := 2x3 3 +5x 3 2 x 4 +x 3 x x x 3 2 x 5 26x 3 x 4 x 5 11x4 2 x 5 +47x 3 x x 4x5 2 +5x 5 3, C 2 := 2x0 3 x 0 2 x 1 2x 0 x1 2 x x 0 2 x 2 +10x 0 x 1 x 2 +x1 2 x 2 11x 0 x2 2 18x 1x2 2 4x 2 3 The cubic fourfold Y is defined by 3C 1 +C 2 +9 (suitable element of Q 1,Q 2,Q 3 ) which has the property that Y (Z/9Z) =!
25 The cubic fourfold The cubic fourfold Y is defined by 2x0 3 x 0 2 x 1 2x 0 x1 2 x x 0 2 x 2 +10x 0 x 1 x 2 +x1 2 x 2 11x 0 x2 2 18x 1x2 2 4x x 0 2 x 3 +18x 0 x 1 x 3 +9x1 2 x 3 +18x 0 x 2 x 3 +18x 1 x 2 x 3 +18x2 2 x 3 +9x 1 x3 2 +6x x0 2 x 4 +9x 0 x 1 x 4 +18x1 2 x 4 9x 0 x 2 x 4 +18x 1 x 2 x 4 +18x2 2 x 4 27x 0 x 3 x 4 +18x 2 x 3 x 4 +15x3 2 x 4 +27x 0 x4 2 36x 2x4 2 +3x 3x x4 3 90x 0 2 x 5 72x 0 x 1 x 5 45x1 2 x 5 18x 1 x 2 x 5 +36x 0 x 3 x 5 45x 1 x 3 x 5 +9x 2 x 3 x 5 60x3 2 x 5 54x 0 x 4 x 5 +27x 1 x 4 x 5 18x 2 x 4 x 5 78x 3 x 4 x 5 33x4 2 x 5 90x 0 x x 3x x 4x x 5 3 = 0.
26 The K3 surface The K3 surface X P(1,1,1,3) is given by w 2 = x x 5 y x 4 y x 3 y x 2 y xy y x 5 z x 4 yz x 3 y 2 z x 2 y 3 z xy 4 z y 5 z x 4 z x 3 yz x 2 y 2 z xy 3 z y 4 z x 3 z x 2 yz xy 2 z y 3 z x 2 z xyz y 2 z xz yz z 6. We show Pic(X ) Z. Hence Br 1 (X )/Br 0 (X ) = 0.
27 Primes of bad reduction of Y 3, 5, 29, 2851, , , , ,
28 Main Theorem Theorem (Berg, V.-A., 2018) There is a degree 2 K3 surface X /Q, together with an A Br(X )[3], such that X (A) and X (A) A =. There is no algebraic Brauer Manin obstruction to the Hasse Principle on X.
Transcendental obstructions on K3 surfaces
Transcendental obstructions on K3 surfaces Anthony Várilly-Alvarado February 28, 2012 1 Introduction Let X be a smooth projective geometrically integral variety over Q. We have an inclusion φ : X(Q) p,
More informationExplicit Arithmetic on Algebraic Surfaces
Explicit Arithmetic on Algebraic Surfaces Anthony Várilly-Alvarado Rice University University of Alberta Colloquium January 30th, 2012 General goal Let X be an algebraic variety defined over Q. Assume
More informationGood reduction of the Brauer Manin obstruction
Good reduction of the Brauer Manin obstruction A joint work in progress with J-L. Colliot-Thélène Imperial College London Schloss Thurnau, July 2010 Notation: k is a number field, k v is the completion
More informationBirational geometry and arithmetic. July 2012
Birational geometry and arithmetic July 2012 Basic questions Let F be a field and X a smooth projective algebraic variety over F. Introduction Basic questions Let F be a field and X a smooth projective
More informationTRANSCENDENTAL OBSTRUCTIONS TO WEAK APPROXIMATION ON GENERAL K3 SURFACES
TRANSCENDENTAL OBSTRUCTIONS TO WEAK APPROXIMATION ON GENERAL K3 SURFACES BRENDAN HASSETT, ANTHONY VÁRILLY-ALVARADO, AND PATRICK VARILLY Abstract. We construct an explicit K3 surface over the field of rational
More informationCUBIC FOURFOLDS FIBERED IN SEXTIC DEL PEZZO SURFACES
CUBIC FOURFOLDS FIBERED IN SEXTIC DEL PEZZO SURFACES NICOLAS ADDINGTON, BRENDAN HASSETT, YURI TSCHINKEL, AND ANTHONY VÁRILLY-ALVARADO Abstract. We exhibit new examples of rational cubic fourfolds. Our
More informationPoint counting and real multiplication on K3 surfaces
Point counting and real multiplication on K3 surfaces Andreas-Stephan Elsenhans Universität Paderborn September 2016 Joint work with J. Jahnel. A.-S. Elsenhans (Universität Paderborn) K3 surfaces September
More informationMORPHISMS TO BRAUER SEVERI VARIETIES, WITH APPLICATIONS TO DEL PEZZO SURFACES
MORPHISMS TO BRAUER SEVERI VARIETIES, WITH APPLICATIONS TO DEL PEZZO SURFACES CHRISTIAN LIEDTKE Abstract. We classify morphisms from proper varieties to Brauer Severi varieties, which generalizes the classical
More informationGeometry dictates arithmetic
Geometry dictates arithmetic Ronald van Luijk February 21, 2013 Utrecht Curves Example. Circle given by x 2 + y 2 = 1 (or projective closure in P 2 ). Curves Example. Circle given by x 2 + y 2 = 1 (or
More informationQuadratic 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 informationSpecial 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 informationRATIONAL POINTS ON SURFACES
RATIONAL POINTS ON SURFACES BIANCA VIRAY ASSISTANT: ARNE SMEETS ARIZONA WINTER SCHOOL 2015 The goal of these lectures is to serve as a user s guide to obstructions to the existence of k-points on smooth
More informationStructure of elliptic curves and addition laws
Structure of elliptic curves and addition laws David R. Kohel Institut de Mathématiques de Luminy Barcelona 9 September 2010 Elliptic curve models We are interested in explicit projective models of elliptic
More information14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski
14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski topology are very large, it is natural to view this as
More informationAlgebra & Number Theory
Algebra & Number Theory Volume 3 2009 No. 7 Cox rings of degree one del Pezzo surfaces Damiano Testa, Anthony Várilly-Alvarado and Mauricio Velasco mathematical sciences publishers ALGEBRA AND NUMBER
More informationVojta s conjecture and level structures on abelian varieties
Vojta s conjecture and level structures on abelian varieties Dan Abramovich, Brown University Joint work with Anthony Várilly-Alvarado and Keerthi Padapusi Pera ICERM workshop on Birational Geometry and
More informationRATIONAL POINTS IN FAMILIES OF VARIETIES
RATIONAL POINTS IN FAMILIES OF VARIETIES MARTIN BRIGHT Contents 1. The Hasse principle 1 2. The Brauer Manin obstruction 4 3. Understanding Brauer groups 7 4. Local solubility in families 11 5. The Brauer
More informationBRAUER MANIN OBSTRUCTIONS TO INTEGRAL POINTS
BRAUER MANIN OBSTRUCTIONS TO INTEGRAL POINTS ANDREW KRESCH AND YURI TSCHINKEL Abstract. We study Brauer Manin obstructions to integral points on open subsets of the projective plane. 1. Introduction Let
More informationON NODAL PRIME FANO THREEFOLDS OF DEGREE 10
ON NODAL PRIME FANO THREEFOLDS OF DEGREE 10 OLIVIER DEBARRE, ATANAS ILIEV, AND LAURENT MANIVEL Abstract. We study the geometry and the period map of nodal complex prime Fano threefolds with index 1 and
More informationThe Brauer group of Kummer surfaces and torsion of elliptic curves
The Brauer group of Kummer surfaces and torsion of elliptic curves Alexei N. Skorobogatov and Yuri G. Zarhin Introduction In this paper we are interested in computing the Brauer group of K3 surfaces. To
More informationCOMPLEX 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 informationTranscendental Brauer elements and descent. elliptic surfaces. Bianca Viray (Brown University)
on elliptic surfaces Bianca Viray Brown University 40 Years and Counting: AWM s Celebration of Women in Mathematics September 17, 2011 The Brauer group Let k be a field of characteristic 0. Definition
More informationFANO VARIETIES OF CUBIC FOURFOLDS CONTAINING A PLANE. 1. Introduction
FANO VARIETIES OF CUBIC FOURFOLDS CONTAINING A PLANE EMANUELE MACRÌ AND PAOLO STELLARI Abstract. We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to
More informationTHE SET OF NONSQUARES IN A NUMBER FIELD IS DIOPHANTINE
THE SET OF NONSQUARES IN A NUMBER FIELD IS DIOPHANTINE BJORN POONEN Abstract. Fix a number field k. We prove that k k 2 is diophantine over k. This is deduced from a theorem that for a nonconstant separable
More informationSPECIAL PRIME FANO FOURFOLDS OF DEGREE 10 AND INDEX Introduction
SPECIAL PRIME FANO FOURFOLDS OF DEGREE 10 AND INDEX 2 OLIVIER DEBARRE, ATANAS ILIEV, AND LAURENT MANIVEL Abstract. We analyze (complex) prime Fano fourfolds of degree 10 and index 2. Mukai gave a complete
More informationThe Brauer group and beyond : a survey
The Brauer group and beyond : a survey Jean-Louis Colliot-Thélène (CNRS et Université Paris-Sud, Paris-Saclay) Summer School Quadratic Forms in Chile 2018 First part : Quadratic forms over function fields
More informationArithmetic applications of Prym varieties in low genus. Nils Bruin (Simon Fraser University), Tübingen, September 28, 2018
Arithmetic applications of Prym varieties in low genus Nils Bruin (Simon Fraser University), Tübingen, September 28, 2018 Background: classifying rational point sets of curves Typical arithmetic geometry
More informationThe set of non-n-th powers in a number field is diophantine Joint work with Jan Van Geel (Gent) Jean-Louis Colliot-Thélène (CNRS et Université
The set of non-n-th powers in a number field is diophantine Joint work with Jan Van Geel (Gent) Jean-Louis Colliot-Thélène (CNRS et Université Paris-Sud, Orsay) Second ERC Research period on Diophantine
More informationa double cover branched along the smooth quadratic line complex
QUADRATIC LINE COMPLEXES OLIVIER DEBARRE Abstract. In this talk, a quadratic line complex is the intersection, in its Plücker embedding, of the Grassmannian of lines in an 4-dimensional projective space
More informationGEOMETRY OF 3-SELMER CLASSES THE ALGEBRAIC GEOMETRY LEARNING SEMINAR 7 MAY 2015
IN GEOMETRY OF 3-SELMER CLASSES THE ALGEBRAIC GEOMETRY LEARNING SEMINAR AT ESSEN 7 MAY 2015 ISHAI DAN-COHEN Abstract. We discuss the geometry of 3-Selmer classes of elliptic curves over a number field,
More informationIsogeny invariance of the BSD conjecture
Isogeny invariance of the BSD conjecture Akshay Venkatesh October 30, 2015 1 Examples The BSD conjecture predicts that for an elliptic curve E over Q with E(Q) of rank r 0, where L (r) (1, E) r! = ( p
More informationA VERY GENERAL QUARTIC DOUBLE FOURFOLD IS NOT STABLY RATIONAL
A VERY GENERAL QUARTIC DOUBLE FOURFOLD IS NOT STABLY RATIONAL BRENDAN HASSETT, ALENA PIRUTKA, AND YURI TSCHINKEL Abstract. We prove that a very general double cover of the projective four-space, ramified
More informationDEGREE AND THE BRAUER-MANIN OBSTRUCTION
DEGREE AND THE BRAUER-MANIN OBSTRUCTION BRENDAN CREUTZ, BIANCA VIRAY, AND AN APPENDIX BY ALEXEI N. SKOROBOGATOV Abstract. Let X P n k is a smooth projective variety of degree d over a number field k and
More informationSOME REMARKS ON BRAUER GROUPS OF K3 SURFACES
SOME REMARKS ON BRAUER GROUPS OF K3 SURFACES BERT VAN GEEMEN arxiv:math/0408006v2 [math.ag] 8 Nov 2004 Abstract. We discuss the geometry of the genus one fibrations associated to an elliptic fibration
More informationK3 Surfaces and Lattice Theory
K3 Surfaces and Lattice Theory Ichiro Shimada Hiroshima University 2014 Aug Singapore 1 / 26 Example Consider two surfaces S + and S in C 3 defined by w 2 (G(x, y) ± 5 H(x, y)) = 1, where G(x, y) := 9
More informationCOMPLEX ALGEBRAIC SURFACES CLASS 15
COMPLEX ALGEBRAIC SURFACES CLASS 15 RAVI VAKIL CONTENTS 1. Every smooth cubic has 27 lines, and is the blow-up of P 2 at 6 points 1 1.1. An alternate approach 4 2. Castelnuovo s Theorem 5 We now know that
More informationArithmetic of del Pezzo surfaces
Arithmetic of del Pezzo surfaces Anthony Várilly-Alvarado Department of Mathematics MS-136 Rice University Houston, TX 77005, USA varilly@rice.edu Introduction These notes were written to accompany a mini-course
More informationAnother way to proceed is to prove that the function field is purely transcendental. Now the coordinate ring is
3. Rational Varieties Definition 3.1. A rational function is a rational map to A 1. The set of all rational functions, denoted K(X), is called the function field. Lemma 3.2. Let X be an irreducible variety.
More informationStrong approximation with Brauer-Manin obstruction for certa. algebraic varieties. Fei XU
Strong approximation with Brauer-Manin obstruction for certain algebraic varieties School of Mathematical Sciences, Capital Normal University, Beijing 100048, P.R.CHINA I. Strong Approximation. Let F be
More informationRational Curves On K3 Surfaces
Rational Curves On K3 Surfaces Jun Li Department of Mathematics Stanford University Conference in honor of Peter Li Overview of the talk The problem: existence of rational curves on a K3 surface The conjecture:
More informationON BRAUER GROUPS OF DOUBLE COVERS OF RULED SURFACES
ON BRAUER GROUPS OF DOUBLE COVERS OF RULED SURFACES BRENDAN CREUTZ AND BIANCA VIRAY Abstract. Let X be a smooth double cover of a geometrically ruled surface defined over a separably closed field of characteristic
More informationThe algebraic Brauer-Manin obstruction on Châtelet surfaces, degree 4 del Pezzo surfaces, and Enriques surfaces. Bianca Lara Viray
The algebraic Brauer-Manin obstruction on Châtelet surfaces, degree 4 del Pezzo surfaces, and Enriques surfaces by Bianca Lara Viray A dissertation submitted in partial satisfaction of the requirements
More informationEFFECTIVITY OF BRAUER MANIN OBSTRUCTIONS ON SURFACES
EFFECTIVITY OF BRAUER MANIN OBSTRUCTIONS ON SURFACES ANDREW KRESCH AND YURI TSCHINKEL Abstract. We study Brauer Manin obstructions to the Hasse principle and to weak approximation on algebraic surfaces
More informationBjorn 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 informationRATIONAL CURVES ON PRIME FANO THREEFOLDS OF INDEX 1
RATIONAL CURVES ON PRIME FANO THREEFOLDS OF INDEX 1 BRIAN LEHMANN AND SHO TANIMOTO Abstract. We study the moduli spaces of rational curves on prime Fano threefolds of index 1. For general threefolds of
More informationArtin-Tate Conjecture, fibered surfaces, and minimal regular proper model
Artin-Tate Conjecture, fibered surfaces, and minimal regular proper model Brian Conrad November 22, 2015 1 Minimal Models of Surfaces 1.1 Notation and setup Let K be a global function field of characteristic
More informationThe 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 informationHeight zeta functions
Geometry Mathematisches Institut July 19, 2006 Geometric background Let X P n be a smooth variety over C. Its main invariants are: Picard group Pic(X ) and Néron-Severi group NS(X ) Λ eff (X ), Λ ample
More informationRational points on diagonal quartic surfaces
Rational points on diagonal quartic surfaces Andreas-Stephan Elsenhans Abstract We searched up to height 10 7 for rational points on diagonal quartic surfaces. The computations fill several gaps in earlier
More informationON PLANAR CREMONA MAPS OF PRIME ORDER
T. de Fernex Nagoya Math. J. Vol. 174 (2004), 1 28 ON PLANAR CREMONA MAPS OF PRIME ORDER TOMMASO DE FERNEX Abstract. This paper contains a new proof of the classification of prime order elements of Bir(P
More informationHYPER-KÄHLER FOURFOLDS FIBERED BY ELLIPTIC PRODUCTS
HYPER-KÄHLER FOURFOLDS FIBERED BY ELLIPTIC PRODUCTS LJUDMILA KAMENOVA Abstract. Every fibration of a projective hyper-kähler fourfold has fibers which are Abelian surfaces. In case the Abelian surface
More informationUNEXPECTED ISOMORPHISMS BETWEEN HYPERKÄHLER FOURFOLDS
UNEXPECTED ISOMORPHISMS BETWEEN HYPERKÄHLER FOURFOLDS OLIVIER DEBARRE Abstract. Using Verbitsky s Torelli theorem, we show the existence of various isomorphisms between certain hyperkähler fourfolds. This
More informationRATIONAL CUBIC FOURFOLDS CONTAINING A PLANE WITH NONTRIVIAL CLIFFORD INVARIANT
RATIONAL CUBIC FOURFOLDS CONTAINING A PLANE WITH NONTRIVIAL CLIFFORD INVARIANT ASHER AUEL, MARCELLO BERNARDARA, MICHELE BOLOGNESI, AND ANTHONY VÁRILLY-ALVARADO Abstract. We isolate a general class of smooth
More informationOLIVIER SERMAN. Theorem 1.1. The moduli space of rank 3 vector bundles over a curve of genus 2 is a local complete intersection.
LOCAL STRUCTURE OF SU C (3) FOR A CURVE OF GENUS 2 OLIVIER SERMAN Abstract. The aim of this note is to give a precise description of the local structure of the moduli space SU C (3) of rank 3 vector bundles
More informationFANO VARIETIES AND EPW SEXTICS
FNO VRIETIES ND EPW SEXTICS OLIVIER DEBRRE bstract. We explore a connection between smooth projective varieties X of dimension n with an ample divisor H such that H n = 10 and K X = (n 2)H and a class
More informationON THE INTEGRAL HODGE CONJECTURE FOR REAL THREEFOLDS. 1. Motivation
ON THE INTEGRAL HODGE CONJECTURE FOR REAL THREEFOLDS OLIVIER WITTENBERG This is joint work with Olivier Benoist. 1.1. Work of Kollár. 1. Motivation Theorem 1.1 (Kollár). If X is a smooth projective (geometrically)
More informationSEMIORTHOGONAL DECOMPOSITIONS AND BIRATIONAL GEOMETRY OF DEL PEZZO SURFACES OVER ARBITRARY FIELDS
SEMIORTHOGONAL DECOMPOSITIONS AND BIRATIONAL GEOMETRY OF DEL PEZZO SURFACES OVER ARBITRARY FIELDS ASHER AUEL AND MARCELLO BERNARDARA Abstract. We study the birational properties of geometrically rational
More informationTHE BRAUER-MANIN OBSTRUCTION FOR INTEGRAL POINTS ON CURVES
THE BRAUER-MANIN OBSTRUCTION FOR INTEGRAL POINTS ON CURVES DAVID HARARI AND JOSÉ FELIPE VOLOCH Abstract. We discuss the question of whether the Brauer-Manin obstruction is the only obstruction to the Hasse
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013
18.78 Introduction to Arithmetic Geometry Fall 013 Lecture #4 1/03/013 4.1 Isogenies of elliptic curves Definition 4.1. Let E 1 /k and E /k be elliptic curves with distinguished rational points O 1 and
More informationChern 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 informationPaolo Stellari TWISTED DERIVED CATEGORIES AND K3 SURFACES
Paolo Stellari TWISTED DERIVED CATEGORIES AND K3 SURFACES Joint with D. Huybrechts: math.ag/0409030 and math.ag/0411541 + S.: math.ag/0602399 + Joint with A. Canonaco: math.ag/0605229 + Joint with D. Huybrechts
More informationNormality of secant varieties
Normality of secant varieties Brooke Ullery Joint Mathematics Meetings January 6, 2016 Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, 2016 1 / 11 Introduction Let X
More informationEXPERIMENTS WITH THE TRANSCENDENTAL BRAUER-MANIN OBSTRUCTION
EXPERIMENTS WITH THE TRANSCENDENTAL BRAUER-MANIN OBSTRUCTION ANDREAS-STEPHAN ELSENHANS AND JÖRG JAHNEL Abstract. We report on our experiments and theoretical investigations concerning weak approximation
More informationOn large Picard groups and the Hasse principle for curves and K3 surfaces. CORAY, Daniel, MANOIL, Constantin
Article On large Picard groups and the Hasse principle for curves and K3 surfaces CORAY, Daniel, MANOIL, Constantin Reference CORAY, Daniel, MANOIL, Constantin. On large Picard groups and the Hasse principle
More informationTORSION AND TAMAGAWA NUMBERS
TORSION AND TAMAGAWA NUMBERS DINO LORENZINI Abstract. Let K be a number field, and let A/K be an abelian variety. Let c denote the product of the Tamagawa numbers of A/K, and let A(K) tors denote the finite
More informationR-EQUIVALENCE ON DEL PEZZO SURFACES OF DEGREE 4 AND CUBIC SURFACES. Zhiyu Tian 1. INTRODUCTION
TAIWANESE JOURNAL OF MATHEMATICS Vol. 19, No. 6, pp. 1603-1612, December 2015 DOI: 10.11650/tjm.19.2015.5351 This paper is available online at http://journal.taiwanmathsoc.org.tw R-EQUIVALENCE ON DEL PEZZO
More informationDel Pezzo Surfaces and the Brauer-Manin Obstruction. Patrick Kenneth Corn. A.B. (Harvard University) 1998
Del Pezzo Surfaces and the Brauer-Manin Obstruction by Patrick Kenneth Corn A.B. (Harvard University) 1998 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor
More informationFANO 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 informationEFFECTIVE BOUNDS FOR BRAUER GROUPS OF KUMMER SURFACES OVER NUMBER FIELDS
EFFECTIVE BOUNDS FOR BRAUER GROUPS OF KUMMER SURFACES OVER NUMBER FIELDS VICTORIA CANTORAL FARFÁN, YUNQING TANG, SHO TANIMOTO, AND ERIK VISSE Abstract We study effective bounds for Brauer groups of Kummer
More informationThen 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 informationDerived categories and rationality of conic bundles
Derived categories and rationality of conic bundles joint work with M.Bernardara June 30, 2011 Fano 3-folds with κ(v ) = V smooth irreducible projective 3-fold over C. V is birational (Reid-Mori, Miyaoka)
More informationOn the computation of the Picard group for K3 surfaces
Under consideration for publication in Math. Proc. Camb. Phil. Soc. 1 On the computation of the Picard group for K3 surfaces By Andreas-Stephan Elsenhans Mathematisches Institut, Universität Bayreuth,
More informationARITHMETIC OF K3 SURFACES DRAFT LECTURE NOTES
ARITHMETIC OF K3 SURFACES DRAFT LECTURE NOTES ANTHONY VÁRILLY-ALVARADO These notes are still under construction! Comments and feedback sent to av15@rice.edu would be most appreciated. Version current as
More informationNon-uniruledness results for spaces of rational curves in hypersurfaces
Non-uniruledness results for spaces of rational curves in hypersurfaces Roya Beheshti Abstract We prove that the sweeping components of the space of smooth rational curves in a smooth hypersurface of degree
More informationEquations for Hilbert modular surfaces
Equations for Hilbert modular surfaces Abhinav Kumar MIT April 24, 2013 Introduction Outline of talk Elliptic curves, moduli spaces, abelian varieties 2/31 Introduction Outline of talk Elliptic curves,
More informationElliptic Curves: An Introduction
Elliptic Curves: An Introduction Adam Block December 206 Introduction The goal of the following paper will be to explain some of the history of and motivation for elliptic curves, to provide examples and
More informationCONSTRUCTION OF NIKULIN CONFIGURATIONS ON SOME KUMMER SURFACES AND APPLICATIONS
CONSTRUCTION OF NIKULIN CONFIGURATIONS ON SOME KUMMER SURFACES AND APPLICATIONS XAVIER ROULLEAU, ALESSANDRA SARTI Abstract. A Nikulin configuration is the data of 16 disjoint smooth rational curves on
More informationFAKE 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 informationHOW 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 informationK3 surfaces of Picard rank one which are double covers of the projective plane
K3 surfaces of Picard rank one which are double covers of the projective plane Andreas-Stephan ELSENHANS a and Jörg JAHNEL a a Universität Göttingen, Mathematisches Institut, Bunsenstraße 3 5, D-37073
More informationPRIME EXCEPTIONAL DIVISORS ON HOLOMORPHIC SYMPLECTIC VARIETIES AND MONODROMY-REFLECTIONS
PRIME EXCEPTIONAL DIVISORS ON HOLOMORPHIC SYMPLECTIC VARIETIES AND MONODROMY-REFLECTIONS EYAL MARKMAN Abstract. Let X be a projective irreducible holomorphic symplectic variety. The second integral cohomology
More informationHyperkähler manifolds
Hyperkähler manifolds Kieran G. O Grady May 20 2 Contents I K3 surfaces 5 1 Examples 7 1.1 The definition..................................... 7 1.2 A list of examples...................................
More informationTHE BRAUER-MANIN OBSTRUCTION AND THE FIBRATION METHOD - LECTURE BY JEAN-LOUIS COLLIOT-THÉLÈNE
THE BRAUER-MANIN OBSTRUCTION AND THE FIBRATION METHOD - LECTURE BY JEAN-LOUIS COLLIOT-THÉLÈNE These are informal notes on the lecture I gave at IU Bremen on July 14th, 2005. Steve Donnelly prepared a preliminary
More informationBEZOUT S THEOREM CHRISTIAN KLEVDAL
BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this
More informationarxiv: v1 [math.ag] 1 Mar 2017
BIRATIONAL GEOMETRY OF THE MODULI SPACE OF PURE SHEAVES ON QUADRIC SURFACE KIRYONG CHUNG AND HAN-BOM MOON arxiv:1703.00230v1 [math.ag] 1 Mar 2017 ABSTRACT. We study birational geometry of the moduli space
More informationKUMMER SURFACES AND K3 SURFACES WITH (Z/2Z) 4 SYMPLECTIC ACTION
KUMMER SURFACES AND K3 SURFACES WITH (Z/2Z) 4 SYMPLECTIC ACTION ALICE GARBAGNATI AND ALESSANDRA SARTI Abstract. In the first part of this paper we give a survey of classical results on Kummer surfaces
More informationRational points on algebraic varieties : a survey
Rational points on algebraic varieties : a survey Jean-Louis Colliot-Thélène (CNRS et Université Paris-Sud, Paris-Saclay) Colloquium, Steklov Institute, Moscow October 6th, 2017 The aim of this talk is
More informationELLIPTIC FIBRATIONS AND SYMPLECTIC AUTOMORPHISMS ON K3 SURFACES
ELLIPTIC FIBRATIONS AND SYMPLECTIC AUTOMORPHISMS ON K3 SURFACES ALICE GARBAGNATI AND ALESSANDRA SARTI Abstract. Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and
More informationCombinatorics and geometry of E 7
Combinatorics and geometry of E 7 Steven Sam University of California, Berkeley September 19, 2012 1/24 Outline Macdonald representations Vinberg representations Root system Weyl group 7 points in P 2
More informationA POSITIVE PROPORTION OF THUE EQUATIONS FAIL THE INTEGRAL HASSE PRINCIPLE
A POSITIVE PROPORTION OF THUE EQUATIONS FAIL THE INTEGRAL HASSE PRINCIPLE SHABNAM AKHTARI AND MANJUL BHARGAVA Abstract. For any nonzero h Z, we prove that a positive proportion of integral binary cubic
More informationK3 surfaces of Picard rank one and degree two
K3 surfaces of Picard rank one and degree two Andreas-Stephan Elsenhans and Jörg Jahnel Universität Göttingen, Mathematisches Institut, Bunsenstraße 3 5, D-37073 Göttingen, Germany elsenhan@uni-math.gwdg.de,
More informationarxiv: v1 [math.ag] 29 Jan 2015
CLASSIFICATIONS OF ELLIPTIC FIBRATIONS OF A SINGULAR K3 SURFACE arxiv:50.0484v [math.ag] 29 Jan 205 MARIE JOSÉ BERTIN, ALICE GARBAGNATI, RUTHI HORTSCH, ODILE LECACHEUX, MAKIKO MASE, CECÍLIA SALGADO, AND
More informationDescent theory for strong approximation for varieties containing a torsor under a torus
Université Paris Sud Faculté des Sciences d Orsay Département de Mathématiques M2 Arithmétique, Analyse, Géométrie Mémoire Master 2 presented by Marco D Addezio Descent theory for strong approximation
More informationEFFECTIVE BOUNDS FOR BRAUER GROUPS OF KUMMER SURFACES OVER NUMBER FIELDS
EFFECTIVE BOUNDS FOR BRAUER GROUPS OF KUMMER SURFACES OVER NUMBER FIELDS VICTORIA CANTORAL FARFÁN, YUNQING TANG, SHO TANIMOTO, AND ERIK VISSE Abstract. We study effective bounds for Brauer groups of Kummer
More informationThe descent-fibration method for integral points - St petersburg lecture
The descent-fibration method for integral points - St petersburg lecture Yonatan Harpaz June 5, 015 1 Introduction Let k be a number field and S a finite set of places of k. By an O S -variety we understand
More informationLogarithmic geometry and rational curves
Logarithmic geometry and rational curves Summer School 2015 of the IRTG Moduli and Automorphic Forms Siena, Italy Dan Abramovich Brown University August 24-28, 2015 Abramovich (Brown) Logarithmic geometry
More informationReciprocity laws and integral solutions of polynomial equations
Reciprocity laws and integral solutions of polynomial equations Jean-Louis Colliot-Thélène CNRS, Université Paris-Sud Clay Mathematical Institute, MSRI Congruences, local fields Let f (x 1,, x n ) be a
More informationRational Points on Curves in Practice. Michael Stoll Universität Bayreuth Journées Algophantiennes Bordelaises Université de Bordeaux June 8, 2017
Rational Points on Curves in Practice Michael Stoll Universität Bayreuth Journées Algophantiennes Bordelaises Université de Bordeaux June 8, 2017 The Problem Let C be a smooth projective and geometrically
More informationarxiv: v3 [math.ag] 25 Mar 2013
NEF AND SEMIAMPLE DIVISORS ON RATIONAL SURFACES arxiv:1104.4270v3 [math.ag] 25 Mar 2013 ANTONIO LAFACE AND DAMIANO TESTA Abstract. In this paper we study smooth projective rational surfaces, defined over
More informationLECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL
LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for non-stable-rationality based on the decomposition of the diagonal in the Chow group. This criterion
More information