The Calculations of 3-Dimensional Ising Model of Finite Width
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NIT ATHEMATICS IN ONSTRUCTION AND THE UILT NVIRONMENT Unit 3:Mathematics in Construction andthe Built EnvironmentNationalNQF Level 3: BTECGuided learning hours: 60Unit abstractConstruction, civil engineering and building services engineering are technical disciplines which require the collection, processing and use of numerical data. For example, in a simple construction project, the dimensions of a structure are designed and specified by the architect or engineer, the cost of the work is determined by the cost control surveyor, the quantities of materials to be ordered are determined by the buyer, and the setting out dimensions and angles may be calculated by the contractor. In more complex situations, design engineers use various formulae to calculate properties such as the rate of flow of water through pipes for drainage calculations, or the levels of bending moments in beams for sizing structural elements.It is therefore essential that learners develop an acceptable understanding of the mathematical methods and techniques required for these key activities, and of how to apply them correctly.The unit explores the rules for manipulation of formulae and equations, calculation of lengths, areas and volumes, determination of trigonometric and geometric properties, and the application of graphical and statistical techniques.Upon completion learners will be able to select and apply appropriate mathematical techniques to address a wide variety of standard, practical, industry-related problems.Learning outcomesOn completion of this unit a learner should:1 Know the basic underpinning mathematical techniques and methods used tomanipulate and/or solve formulae, equations and algebraic expressions2 Be able to select and correctly apply mathematical techniques to solve practicalconstruction problems involving perimeters, areas and volumes3 Be able to select and correctly apply a variety of geometric and trigonometrictechniques to solve practical construction problems4 Be able to select and correctly apply a variety of graphical and statisticaltechniques to solve practical construction problems.NIT ATHEMATICS IN ONSTRUCTION AND THE UILT NVIRONMENTUnit content1Know the basic underpinning mathematical techniques and methods used to manipulate and/or solve formulae, equations and algebraic expressionsMathematical techniques and methods: mathematical operators; factorization;expansion; transposition; substitution and elimination; rounding; decimal places;significant figures; approximation; truncation errors and accuracy; calculatorfunctions and useFormulae, equations and algebraic expressions: linear; simultaneous; andquadratic equations; arithmetic progressions; binomial theorem2 Be able to select and correctly apply mathematical techniques to solvepractical construction problems involving perimeters, areas and volumesPerimeters, areas and volumes: calculations both for simple and compoundshapes, eg rectangles, trapeziums, triangles, prisms, circles, spheres, pyramids, cones and both regular and irregular surface areas and volumesMathematical techniques: simple mensuration formulae and numerical integration methods (mid-ordinate rule; trapezoidal rule; Simpson’s rule)3Be able to select and correctly apply a variety of geometric and trigonometric techniques to solve practical construction problemsGeometric techniques: properties of points, lines, angles, curves and planes;Pythagoras’ rule; radians; arc lengths and areas of sectors.Trigonometric techniques: sine, cosine, tangent ratios; sine rule; cosine rule;triangle area rules4Be able to select and correctly apply a variety of graphical and statistical techniques to solve practical construction problemsGraphical techniques: Cartesian and Polar co-ordinates; intersections of graph lines with axes; gradients of straight lines and curves; equations of graphs; areas under graphs; solution of simultaneous and quadratic equationsStatistical techniques: processing large groups of data to achieve mean, median, mode and standard deviation; cumulative frequency, quartiles, quartile range;methods of visual presentationU N I T 3: M A T H E M A T I C S I N C O N S T R U C T I O N A N D T H E B U I L T E N V I R O N M E N TE d e x c e l L e v e l 3 B T E C N a t i o n a l s i n C o n s t r u c t i o n – I s s u e 1 – A p r i l 2007 © E d e x c e l L i m i t e d 20073G r a d i n g g r i dI n o r d e r t o p a s s t h i s u n i t , t h e e v i d e n c e t h a t t h e l e a r n e r p r e s e n t s f o r a s s e s s m e n t n e e d s t o d e m o n s t r a t e t h a t t h e y c a n m e e t a l l o f t h e l e a r n i n g o u t c o m e s f o r t h e u n i t . T h e c r i t e r i a f o r a p a s s g r a d e d e s c r i b e t h e l e v e l o f a c h i e v e m e n t r e q u i r e d t o p a s s t h i s u n i t .G r a d i n g c r i t e r i aT o a c h i e v e a p a s s g r a d e t h e e v i d e n c e m u s t s h o w t h a t t h e l e a r n e r i s a b l e t o :T o a c h i e v e a m e r i t g r a d e t h e e v i d e n c e m u s t s h o w t h a t , i n a d d i t i o n t o t h e p a s s c r i t e r i a , t h e l e a r n e r i s a b l e t o :T o a c h i e v e a d i s t i n c t i o n g r a d e t h e e v i d e n c e m u s t s h o w t h a t , i n a d d i t i o n t o t h e p a s s a n d m e r i t c r i t e r i a , t h e l e a r n e r i s a b l e t o :P 1u s e t h e m a i n f u n c t i o n s o f a s c i e n t i f i c c a l c u l a t o r t o p e r f o r m c a l c u l a t i o n s a n d a p p l y m a n u a l c h e c k s t o r e s u l t sM 1 s e l e c t a n d a p p l y a v a r i e t y o f a l g e b r a i c m e t h o d s t o s o l v e l i n e a r , q u a d r a t i c a n d s i m u l t a n e o u s l i n e a r a n d q u a d r a t i c e q u a t i o n sD 1i n d e p e n d e n t l y u n d e r t a k e c h e c k s o n c a l c u l a t i o n s u s i n g r e l e v a n t a l t e r n a t i v e m a t h e m a t i c a l m e t h o d s a n d m a k e a p p r o p r i a t e j u d g m e n t s o n t h e o u t c o m eP 2u s e s t a n d a r d m a t h e m a t i c a l m a n i p u l a t i o n t e c h n i q u e s t o s i m p l i f y e x p r e s s i o n s a n d s o l v e a v a r i e t y o f l i n e a r f o r m u l a e P 3u s e g r a p h i c a l m e t h o d s t o s o l v e l i n e a r a n d q u a d r a t i c e q u a t i o n sM 2 e x t r a c t d a t a , s e l e c t a n d a p p l y a p p r o p r i a t e a l g e b r a i c m e t h o d s t o f i n d l e n g t h s , a n g l e s , a r e a s a n d v o l u m e s f o r o n e 2D a n d o n e 3D c o m p l e x c o n s t r u c t i o n i n d u s t r y r e l a t e d p r o b l e m s P 4p r o d u c e c l e a r a n d a c c u r a t e a n s w e r s t o a v a r i e t y o f p r o b l e m s a s s o c i a t e d w i t h s i m p l e p e r i m e t e r s , a r e a s a n d v o l u m e sP 5p r o d u c e c l e a r a n d a c c u r a t e a n s w e r s t o a v a r i e t y o f s i m p l e 2D t r i g o n o m e t r i c p r o b l e m sM 3 u s e s t a n d a r d d e v i a t i o n t e c h n i q u e s t o c o m p a r e t h e q u a l i t y o f m a n u f a c t u r e d p r o d u c t s u s e d i n t h e c o n s t r u c t i o n i n d u s t r y .D 2i n d e p e n d e n t l y d e m o n s t r a t e a n u n d e r s t a n d i n g o f t h e l i m i t a t i o n s o f c e r t a i n s o l u t i o n s i n t e r m s o f a c c u r a c y , a p p r o x i m a t i o n s a n d r o u n d i n g e r r o r s .U N I T 3: M A T H E M A T I C S I N C O N S T R U C T I O N A N D T H E B U I L T E N V I R O N M E N T4E d e x c e l L e v e l 3 B T E C N a t i o n a l s i n C o n s t r u c t i o n – I s s u e 1 – A p r i l 2007 © E d e x c e l L i m i t e d 2007G r a d i n g c r i t e r i aT o a c h i e v e a p a s s g r a d e t h e e v i d e n c e m u s t s h o w t h a t t h e l e a r n e r i s a b l e t o :T o a c h i e v e a m e r i t g r a d e t h e e v i d e n c e m u s t s h o w t h a t , i n a d d i t i o n t o t h e p a s s c r i t e r i a , t h e l e a r n e r i s a b l e t o :T o a c h i e v e a d i s t i n c t i o n g r a d e t h e e v i d e n c e m u s t s h o w t h a t , i n a d d i t i o n t o t h e p a s s a n d m e r i t c r i t e r i a , t h e l e a r n e r i s a b l e t o :P 6p r o d u c e c l e a r a n d a c c u r a t e a n s w e r s t o a v a r i e t y o f s i m p l e g e o m e t r i c p r o b l e m sP 7d e s c r i b e a n d i l l u s t r a t e t h e u s e o f s t a t i s t i c s i n t h e c o n s t r u c t i o n i n d u s t r y .NIT ATHEMATICS IN ONSTRUCTION AND THE UILT NVIRONMENT Essential guidance for tutorsDeliveryIt is important that learners possess the basic tool kit for simplifying and solving a variety of formulae and expressions. Practical mathematics has at its core some fundamental techniques and methods which must become second nature to the learner. To achieve this the learners need time to follow through worked examples under the guidance of the tutor and then practise this work at their own pace. The work should not be seen as a rote memory exercise but as the application of basic rules as part of a structured and logical procedure. In this way they will be able to cope with a variety of numerical problems that come their way in the course of their studies and later on during their professional career.Learning outcome 1 forms the basis for all the following outcomes and should therefore be covered first. The following two learning outcomes reflect the application of important mathematical skills and techniques in the solution of spatial problems using mensuration, geometry and trigonometric methods and techniques. The final learning outcome covers the separate topic of statistics and their presentation, analysis and interpretation. This structure would therefore indicate at least three assessment instruments.Teaching and learning strategies designed to support delivery of this unit should involve theory, worked examples and then, most importantly, practice. Practice is the key word and the learners must be given many opportunities to practise the relevant techniques. The use of formative tests and coursework will help the learner to see where they may be going wrong. Within the scheme of work there should be time allowed for regular workshops and/or tutorials to reinforce the learning process. It would also be beneficial to provide additional support and tutoring for the weaker learner through the provision of qualified classroom assistants or other forms of learning support. Delivery should stimulate, motivate, educate and enthuse the learner.It is anticipated that this unit will be delivered in the first year of the programme to enable an early foundation to be established for the technical and numerically-based units that are to follow.Group activities are permissible, but tutors will need to ensure that individual learners are provided with equal experiential and assessment opportunities. Health, safety and welfare issues are paramount and should be strictly reinforced through close supervision of all workshops and activity areas, and risk assessments must be undertaken prior to practical activities. Centres are advised to read the Delivery approach section on page 24, and Annexe G: Provision and Use of Work Equipment Regulations 1998 (PUWER).NIT ATHEMATICS IN ONSTRUCTION AND THE UILT NVIRONMENTAssessmentEvidence for this unit may be gathered from short time-controlled phase tests, tutor-provided practical construction scenarios, case studies, practical work or traditional example-based methods.There are many suitable forms of assessment that could be employed. Some examples of possible assessment approaches are suggested below. However, these are not intended to be prescriptive or restrictive, and are provided as an illustration of the alternative forms of assessment evidence that would be acceptable. General guidance on the design of suitable assignments is available on page 19 of this specification. Some criteria can be assessed directly by the tutor during practical activities. If this approach is used, suitable evidence would be observation records or witness statements. Guidance on the use of these is provided on the Edexcel website.A variety of assessment instruments should be used. For the earlier work involving the basic rules of algebra it is suggested that a short time-controlled assignment is given with some degree of revision coaching provided. This will stimulate knowledge and understanding of the basic techniques and methods, as well as developing mental agility. The assessments involving applied mensuration, geometry and trigonometry could be written into practical scenario-based problems that reflect the vocational pathway being studied. The final section on statistics could be in the form of a seminar or oral presentation including the production of visual aids and the use of spreadsheets. There could also be useful opportunities for self- and peer-assessment in this type of situation, but this would need to be carefully balanced against tutor assessment to ensure the validity of the evidence.To achieve a pass grade learners must meet the seven pass criteria listed in the grading grid.For P1, learners must be able to use the main functions of a scientific calculator with confidence and efficiency and be able to produce rough mental and manual checks on the answers achieved. They should give their answers in the appropriate form taking into account truncation, rounding and standard form. In all industry-related problems the correct units should be used.For P2, learners must set out the solutions using the correct mathematical conventions. All solutions to formulae should be re-substituted back to check answers. Minor oversights are acceptable when simplifying expressions provided that the majority of the work and methods are correct.For P3, for simple linear solutions learners will be expected to plot graphs by appropriate selection of a range of x-variables. For more complex types such as simultaneous equation or those including powers the range of values can be provided. All graphs should be correctly annotated and labelled.For P4, learners should provide solutions which clearly show how they have approached the mensuration problem and collated the data, eg the appropriate use of labelled diagrams. The solutions should be set out methodically and clearly using the correct mathematical conventions. Units should be clearly stated for the problems involving physical properties.NIT ATHEMATICS IN ONSTRUCTION AND THE UILT NVIRONMENT For P5, learners should provide solutions which clearly show how they have approached the trigonometric problem and collated the data, eg the appropriate use of labelled diagrams. The solutions should be set out methodically and clearly using the correct mathematical conventions. Units should be clearly stated for the problems involving physical properties.For P6, learners should provide solutions which clearly show how they have approached the geometric problem and collated the data, for example the appropriate use of labelled diagrams. The solutions should be set out methodically and clearly using the correct mathematical conventions. Units should be clearly stated for the problems involving physical properties.For P7, learners need to demonstrate how industry-related data is calculated and presented. The calculation of values and their representation can be integrated within spreadsheet work. Learners should interpret the results and draw relevant conclusions. To achieve a merit grade learners must meet all of the pass criteria and the three merit grade criteria.For M1, learners, with minimal tutor support, should demonstrate how to solve linear, quadratic and simultaneous linear and quadratic equations equations using solution by: formula, by factorisation and by the ‘perfect squares’ method. The structure and layout of the solutions should show a correct and methodical progression through the various stages of the calculations.For M2, learners, with minimal tutor support, should be able to extract data from complex industry-related problems: one 2D and one 3D. They should present and apply the data to suitable mathematical models and perform the necessary calculations. The solutions should be set out methodically and clearly using the correct mathematical conventions. The units should be clearly stated throughout. For M3, learners, with minimal tutor support, should be able to use standard deviation techniques to compare and comment on the material properties of manufactured products, eg cube test strength and steel tensile strength. Access to secondary research data will be sufficient to cover this criterion.To achieve a distinction grade learners must meet all of the pass criteria and merit grade criteria and the two distinction grade criteria.For D1, learners should independently undertake alternative mathematical methods to check solutions, using appropriate and relevant techniques A high level of clarity and presentation should be displayed. For example, this could include the learner independently developing and using spreadsheets for complex multi-stage calculation to confirm manually-generated results. Learners would also be expected to draw suitable conclusions from the resultant outcomes that relate to industrial situations. For D2, learners should independently demonstrate an understanding of the accuracy and rounding of data, and its effect on calculated outcomes. This includes the application of binomial theory to small errors. Suitable conclusions should be made on the resultant outcomes, in an industrial context.NIT ATHEMATICS IN ONSTRUCTION AND THE UILT NVIRONMENTLinks to National Occupational Standards, other BTEC units, other BTEC qualifications and other relevant units and qualificationsIt is anticipated that this unit will be delivered within the early stages of the programme. This will enable the learner to apply the underpinning knowledge, skills and understanding gained in this unit to the study of other, more specialised units. This unit builds upon the knowledge, understanding and skills gained through the Mathematics taught at Key Stage 4, and underpins progression to the numerically-based units within the BTEC National Diploma, including: Unit 4: Science and Materials in Construction and the Built Environme nt; Unit 13: Environmental Science in Construction; Unit 14: Structural Mechanics in Construction and Civil Engineering, Unit 19: Further Mathematics in Construction and the Built Environment, and various building services units.The unit has no direct mapping links to the CIC Occupational Standards at Level 3. The unit provides opportunities to gain Level 3 key skills in: application of number; and information and communication technology. Opportunities for satisfying requirements for Wider Curriculum Mapping are summarised in Annexe F: Wider curriculum mapping.Essential resourcesThe application of mathematical techniques requires little in the way of resources other than scientific calculators and drawing equipment. Both of these are implicit requirements of many other units and, therefore, no additional extra resources are required for this unit, other than a range of industry-contextualised, realistic and feasible project material appropriate to the application of a range of mathematical methods.Where spreadsheets are incorporated into the delivery or assessment scheme, learners should be provided with the appropriate access to suitable software. Indicative reading for learnersTextbooksBird and May — Technician Mathematics 2, 3rd Edition (Pearman, 1994)ISBN 0582234271Greer and Taylor — BTEC National NII: Mathematics for Technicians (Nelson Thornes, 1994) ISBN 0748717013Tourret A — Applying Maths in Construction: Student Book (Architectural Press, 1997) ISBN 0340652950Tourret A and Humphreys — Applying Maths in Construction: Teacher’s Pack (Architectural Press, 1997) ISBN 0340652969NIT ATHEMATICS IN ONSTRUCTION AND THE UILT NVIRONMENT Key skillsAchievement of key skills is not a requirement of this qualification but it is encouraged. Suggestions of opportunities for the generation of Level 3 key skill evidence are given here. Staff should check that learners have produced all the evidence required by part B of the key skills specifications when assessing this evidence. Learners may need to develop additional evidence elsewhere to fully meet the requirements of the key skills specifications.Application of number Level 3When learners are:They should be able to develop the followingkey skills evidence:•performing standardcalculations associated withmensuration, trigonometry orgeometry•selecting appropriateformulae to solve industry-related problems•using appropriate formulae to solve industry-relatedproblems•interpreting, presenting and justifying the results ofcalculations. N3.1 Plan an activity and get relevantinformation from relevant sources.N3.2 Use your information to carry out multi-stage calculations to do with:a amounts or sizesb scales or proportionc handling statisticsd using formulae.N3.3 Interpret the results of your calculations, present your findings and justify yourmethods.Information and communication technology Level 3When learners are:They should be able to develop the followingkey skills evidence:•researching industry data to analyse statistical trends andmeans•manipulating and presenting data using spreadsheets •undertaking checks onmanual calculations bydeveloping and usingrelevant spreadsheets. ICT3.1 Search for information, using different sources, and multiple search criteria in atleast one case.ICT3.2 Enter and develop the information and derive new information.ICT3.3 Present combined information such as text with image, text with number,image with number.。
a rX iv:mat h /5716v1[mat h.AG ]1J ul25The abelian fibration on the Hilbert cube of a K3surface of genus 9Atanas Iliev and Kristian Ranestad Abstract In this paper we construct an abelian fibration over P 3on the Hilbert cube of the primitive K3surface of genus 9.After the abelian fibration constructed by Mukai on the Hilbert square on the primitive K3surface of genus 5,this is the second example where the abelian fibration on such H ilb n S is directly constructed.Our example is also the first known abelian fibration on a Hilbert scheme H ilb n S of a primitive K3surface S which is not the Hilbert square of S ;the primitive K 3surfaces on the Hilbert square of which such a fibration exists are known by a recent result of Hassett and Tschinkel.1Introduction Generalities .The smooth complex projective variety X is a hyperk¨a hler manifold if X is simply connected and H 0(Ω2X )=C ωfor an everywhere non-degenerate form ω,the symplectic form on X .In particular,by the non-degeneracy of ω,a hyperk¨a hler manifold is always even-dimensional and with a trivial canonical class,see e.g.[4]or [5]for a survey of the basic properties of hyperk¨a hler manifolds.A fibre space structure,or a fibration on the smooth projective manifold X is a projective morphism,with connected equidimensional fibers,from X onto a normal projective variety Y such that 0<dim (Y )<dim (X ).By the theorems of Matsushita (see [10]),any fibration f :X →Y on a hyperk¨a hler 2n -fold X is always a Lagrangianabelian fibration,i.e.the general fiber F y =f −1(y )⊂X must be an abelian n -fold which is also a Lagrangian submanifold of X with respect to the form ω;in addition Y has to be a Fano n -fold with the same Betti numbers as P n .The question when a hyperk¨a hler 2n -fold X admits a structure of a fibration in the above sense is among the basic problems for hyperk¨a hler manifolds (cf.[4],p.171).An additional question is whether the base Y of any such fibration is always the projective space P n ,ibid.For a more detailed discussion on the problem,see the paper [16]of J.Sawon.As shown by Beauville,the Hilbert schemes H ilb n S of length n zero-subschemes of smooth K3surfaces S are hyperk¨a hler manifolds,see[1].In the special case when S is an elliptic K3surface,the ellipticfibration S→P1 on S induces naturally a structure of an abelianfibration H ilb n S→P n for any n≥2, see e.g.[16],Ex.3.5.Another candidate is Beauville’sfibration H ilb g S···>P g for a K3surface S containing a smooth curve of genus g≥2,in particular if S=S2g−2is a primitive K3surface of genus g≥2,ibid.Ex.3.6.But Beauville’sfibration is not regular,i.e.it is not afibration in the above sense.Until now,the only abelianfibration constructed directly on a Hilbert power of a primitive K3surface S=S2g−2,g≥2is the abelianfibration due to Mukai([11]) on the Hilbert square of the primitive K3surface S8of genus5.On the base of this example,and by using a deformation argument,Hassett and Tschinkel manage to prove the existence of an abelianfibration over P2on the Hilbert square of the primitive K3 surface of degree2m2,for any m≥2,see[3].The main result of the paper and a conjecture.In this paper we construct an abelianfibration over P3on the Hilbert cube of the general primitive K3surface of genus9.After Mukai’s example,this is the second case where the abelianfibration on H ilb n S is constructed directly.This is also thefirst known abelianfibration on a Hilbert power of a primitive K3surface,which is not a Hilbert square.In addition,we consider the conjecture that if S is a primitive K3surface of genus g≥2,then the Hilbert scheme X=H ilb n S admits a structure of an abelianfibration if and only if2g−2=m2(2n−2)for some integer m≥2.This conjecture is posed as a question by several authors(cf.[4],[3]),so we claim no originality.The Hassett-Tschinkel result covers the cases n=2,m≥2of this conjecture,and Mukai’s example corresponds to the particular case n=m=2.Our example proves the conjecture in the case n=3,m=2.The construction of the abelianfibration in brief.To explain our construc-tion,we compare it with Mukai’s construction of the abelianfibration on the Hilbert square of the primitive K3surface S8of genus5.The general S=S8is a complete intersection of three quadrics in P5.Let P(H0(I S(2)))=ˇP2S be the plane of quadratic equations of S in P5.InsideˇP2S the subset∆S of singular quadrics is a smooth plane sextic,and the double covering ofˇP2S branched along∆S defines uniquely a K3surface F S of genus2,the dual K3surface of S(cf.[14]).Any subschemeξ∈H ilb2S spans a line lξ=<ξ>⊂P5,and the set(2))of quadrics containing S and lξis a line inˇP2S.If P2S is the dual Lξ=P(H0(I S∪lξplane ofˇP2S,the association f:ξ→Lξdefines a regular mapf:H ilb2S→P2S.In turn,afixed line L∈P2S defines uniquely a3-fold X L⊃S in P5as the common zero locus of the quadrics Q∈L⊂ˇP2S.By the definition of f,the0-schemesξin thepreimage A L=f−1(L)are intersected on S by the lines L⊂X L,thus thefiber A L is isomorphic to the Fano family F(X L)of lines on the threefold X L.The theorems in [10]imply that the map f is an abelianfibration,in particular for the general L the family F(X L)is an abelian surface.The last is classically known:the general X L is the quadratic complex of lines,and the family of lines F(X L)is the jacobian of a genus 2curve F L,the dual curve to X L;the last turns out to be a hyperplane section of F S –the double covering F L of L branched along the6-tuple∆L=∆S∩L,see e.g.[2].Let now S=S16⊂P9be a general primitive K3surface of genus9,and let H ilb3S be the Hilbert scheme of length3zero-schemesξ⊂S.Following the above example, we shall define step-by-step a mapf:H ilb3S→P3.The analog of the projective space P5⊃S8is now the Lagrangian grassmannian LG(3,6),the closed orbit of the projectivized irreducible14-dimensional representation V14of the symplectic group Sp(3).The Lagrangian grassmannianΣ=LG(3,6)is a smooth6-fold of degree16in P13=P(V14),and by the classification results of Mukai [12],the general primitive K3surface S=S16is a linear section of the Lagrangian grassmannian LG(3,6)⊂P13=P(V14)by a codimension3subspace P9identified with Span(S).The projectivized dual representation V∗14of Sp(3)has an open orbit inˆP13= P(V∗14),and the complement to this orbit is a quartic hypersurfaceˆF.The space of linear equationsˆP3S=P9,⊥Sof S⊂LG(3,6)is a projective3-space inˆP13=P(V∗14), which intersectsˆF along a smooth quartic surface F S⊂ˆP3S,the Sp(3)-dual quartic ofS,see[8].The base of f is now the dual3-space P3S=ˆP3,∗S ,interpreted equivalentlyas the set of all3-folds X h=Σ∩P10h,h∈P3S which lie insideΣ=LG(3,6)and pass through S.Structure of the paper.The crucial step in our construction is tofind the analog of lines through2points on S8.It turns out that these are twisted cubics on Σ=LG(3,6),and the crucial property of these cubics is:Through the general tripleξof points onΣpasses a unique twisted cubic Cξthat lies inΣ.In Sections2and3we show that on the general K3surface S=S16⊂LG(3,6),if ξis a0-scheme on S,then there exists on LG(3,6)a unique connected rational cubic curve Cξintersecting on S the zero-schemeξ.This identifies H ilb3S with the6-fold Hilbert scheme C(S)of twisted cubic curves C⊂LG(3,6)that lie in Fano3-folds X h,h∈P3S.If F S⊂ˆP3S is the Sp(3)-dual quartic surface of S,any h∈P3S=ˆP3,∗S defines ahyperplane section F h⊂F S,the dual plane quartic of X h.For the general h,F h is a smooth plane quartic and X h is a smooth prime Fano3-fold of genus9;one can regard this X h as the analog of the general quadratic complex of lines X L through the K3 surface S8.For h′=h′′the families of cubics C(X h′)and C(X h′′)do not intersect eachother,thus giving a regular mapf:H ilb3S∼=C(S)→P3S,with h=f(ξ)identified with the10-space P10ξ=Span(S∪Cξ).The results of[10] imply that f is an abelianfibration,and the construction of f identifies thefiber A h=f−1(h)of f with the Hilbert scheme C(X h)of twisted cubic curves on the3-fold X h.In Section4we show that for the general h the family C(X h)is nothing else but the jacobian J(F h)of the dual plane quartic F h to the3-fold X h.Just as in the case for the quadratic complex of lines,the abelian threefold A h=C(X h)can be identified with the intermediate jacobian J(X h)of the Fano3-fold X h.At the end,in Section5we describe the group law on the generalfiber A h of f,in the interpretation of A h as the Hilbert scheme C(X h)of twisted cubic curves on X h. More precisely,any twisted cubic C o⊂X h defines on the abelian3-fold A h=C(X h)an additive group structure with C o=0,and we identify which cubic curve on X=X h is the sum under this group operation of two general cubics C′,C′′⊂X.Notice that this is the analog of the Donagi’s group law on the family F(X L)of lines on the3-fold quadratic complex of lines(see[2]),identified with the generalfiber of Mukai’s abelian fibration on H ilb2S8.We thank Y.Prokhorov for the advice in computing the multiplicity coefficients in Proposition4.5,and S.Popescu and D.Markushevich for the helpful discussions.2Twisted cubic curves on LG(3,6)Let V be6-dimensional vector space,letα∈∧2V∗be a nondegenerate2-form on V. The natural linear map dα:∧3V→V induced byαhas a14-dimensional kernel W= ker dα.Consider the Pl¨u cker embedding G(2,V)⊂P(∧3V).ThenΣ=LG(3,V)= P(W)∩G(2,V)⊂P(∧3V)is the6-dimensional Grassmannian of Lagrangian planes in P5=P(V)with respect toα.If H⊂Σis the hyperplane divisor in this Pl¨u cker embedding,then the degree ofΣis H6=16,while the canonical divisor KΣ=−4H. Thus a general linear section P10∩Σ⊂P(W)is a Fano threefold of genus9,while a general linear section P9∩Σ⊂P(W)is a K3surface of genus9.P roposition2.1.(Mukai[12])A general prime Fano threefold of genus9is the linear section P10∩LG(3,V)for some P10⊂P(W).A general K3-surface of genus9is the linear section P9∩LG(3,V)for some P9⊂P(W).Let Sp(3)⊂SL(V)be the symplectic group of linear transformations that leaves αinvariant.Then W is an irreducible Sp(3)-representation and the four orbit closures of Sp(3)on P(W)areΣ⊂Ω⊂F⊂P(W)Their dimensions are6,9,12and13.In particular F is a quartic hypersurface,the union of projective tangent spaces toΣ.Similarly,there are four orbit closures of Sp(3)in the dual space:Σ∗⊂Ω∗⊂F∗⊂P(W∗),isomorphic to the orbit closures in P(W).In particular,the quartic hypersurface F∗is the dual variety ofΣand parametrizes tangent hyperplane sections toΣ.If X=Σ∩L is a linear section ofΣ,and L⊥⊂P(W∗)is the orthogonal linear space,then we denote by F X the intersection L⊥∩F∗and call it the Sp(3)-dual variety to X.The four orbits in P(W)are characterized by secant properties ofΣ(cf.[8]):P roposition2.2.Letω∈P(W),then:(a)Ifω∈P(W)−Ω,then throughωpasses a unique bisecant or tangent line lωto Σ.The line lωis tangent toΣif and only ifω∈F.(b)Ifω∈Ω−Σthen the set of lines which pass throughωand are bisecant or tangent toΣsweep out a4-space P4ω⊂P(W),and the intersection Qω=P4ω∩Σis a smooth3-fold quadric.In the space P5,there exists a point x=x(ω)such that the quadric Qω=Q x(ω)⊂Σcoincides with the set of Lagrangian planes that pass through the point x(ω).FurthermoreL emma2.3.(a)A line not contained inΣintersectΣin a0-scheme of length≤2.(b)Σdoes not contain planes,and a plane that intersectsΣalong a conic section is contained inΩ.(c)a plane P∼=P2such that the intersection scheme P∩Σcontains a0-scheme Z of length3either intersectsΣexactly at Z or P∩Σcontains a line or a conic.(d)A3-space P∼=P3such that the intersection P∩Σcontains a curve C of degree3either intersectsΣexactly along C and C is defined by a determinantal net of quadrics or P⊂Ωand P∩Σis a quadric surface of rank≥3.Proof.SinceΣis defined by quadrics,(a)follows.The planes that are parameterized by a conic have a common point p∈P5,so the conic lies in the3-dimensional smooth quadric Q p(2.2).Since this quadric is smooth,it and therefore alsoΣ,contains no planes,and the conic is a plane section of Q p.Hence P⊂<Q p>⊂Ωand(b)follows. For(c),consider a plane P whose intersection withΣcontains a scheme Z of length 4,and assume that the intersection is zero-dimensional.Then Z must be a complete intersection of two conics,and the intersection P∩Σis precisely Z.Let S be a general P9that contains P,and let S=Σ∩P9.Then,by Bertini,S is a smooth surface.In fact S is a K3-surface.Since Z define dependent conditions on hyperplanes h,there is a nontrivial extension of I Z,S by O S which define a rank2sheaf E on S.Since no length three subscheme on S is contained in a line,E is a vector bundle(cf.[15]),and since no plane intersects S in a scheme of lengthfive E is base point free.Therefore a general section of E is a smooth subscheme of length four contained in a plane.But throughthis subscheme there are two lines that meet in a point outside S,contradicting the above proposition.Let P be a P3that intersectsΣin a curve C.By(c)this curve has degree at most 3,so for(d)we may assume that the degree is three.Again by(c),the intersection is pure:There are no zero-dimensional components.Now,C cannot be a plane curve by part a).Furthermore,sinceΣis the intersection of quadrics,C is contained in at least three independent quadrics.Pick two without common component,then for degree reasons alone they link C to a line and C is defined by a determinantal net of quadrics. If P intersectsΣin a surface,this surface,by(a)is a quadric of rank at least3,and by(b)lies inΩ.L emma2.4.Let M be a2×3-matrix of linear forms in P3whose rank1locus is a curve C M of degree3,and let p∈P3\C M.Then there is a unique line passing through p that intersects C M in a scheme of length2,unless p lies in a plane that intersects C M in a curve of degree2.Proof.Let M(p)be M evaluated in p.Then M(p)has rank2,so let(a1,a2,a3)be the unique(up to scalar)solution to M(p)·a=0.Then M·a defines two linear forms that vanish on p.If they are independent,they define a line that intersect C M in a scheme of length2,and if they are dependent,then they define a plane that intersect C M in a curve of degree2.The uniqueness of the line in thefirst case follows by construction.L emma2.5.Let P9⊂P(W)be general.In particular assume that S=Σ∩P9is a smooth K3surface of genus9with no rational curve of degree less than4.Then:(a)A line l can intersect S in at most a0-scheme of length≤2.(b)If a plane P2intersects S in a scheme containing a0-schemeξof length3then P2=<ξ>and P2∩S=ξ.(c)If a3-space P3is such that the intersection scheme P3∩S contains a0-schemeξof length3and the intersection P3∩Σcontains a curve C of degree3,then P3∩P9=<ξ>,P3∩S=ξand P3∩Σ=C;here P9=<S>.Proof.(a)follows immediately from Lemma2.3.By(a),<ξ>=P2.If the intersection scheme Z=P2∩S⊃ξcontainsξproperly,then by Lemma2.3Z will contain a line or a conic.But S contains no curves of degree less than4,so(b)follows.(c)Since P3∩S⊃ξthen P3contains the plane P2ξ=<ξ>⊂P9,by(b).Therefore either P3⊂P9=<S>or P3∩P9=<ξ>.But if P3⊂P9then the curve C⊂P3∩Σwill be contained in S=Σ∩P9,which is impossible since by assumption S=S16does not contain curves.of degree3.Therefore P3∩P9=<ξ>and P3∩S=<ξ>∩S=ξ.If P3∩Σcontains more than the curve C then by Lemma2.3,the intersection Q=P3∩Σwill be a quadric surface of rank at least3.But then S∩Q is a conic, contradicting the assumption that S has no curve of degree less than4.D efinition2.6.Let H ilb3t+1(Σ)be the Hilbert scheme of twisted cubic curves con-tained inΣand let H ilb3(Σ)be the Hilbert scheme of length three subschemes ofΣ.SinceΣcontains no planes,every member C of H ilb3t+1(Σ)is a curve of degree 3defined by a determinantal net of quadrics:C lies in at least3quadrics,and since there are no planes inΣtwo general ones intersect in a curve of degree4that links C to a line.Consider the incidenceI3={(ξ,C)|ξ⊂C}⊂H ilb3(Σ)×H3t+1(Σ)and the restriction to S:I3(S)={(ξ,C)∈I3|ξ⊂S}D efinition2.7.Let C(S)⊂H ilb3t+1(Σ)be the image of the projectionI3(S)→H ilb3t+1(Σ),i.e.the Hilbert scheme of twisted cubic curves inΣthat intersect S in a scheme of length three.C orollary2.8.Let S=P9∩Σ∈P(W)be a smooth linear section with no rational curves of degree less than four.(a)C(S)⊂H ilb3t+1(Σ)is a closed subscheme.(b)The intersection mapσ:C(S)→H ilb3S,C→C∩Sis well defined on any C∈C(S).Proof.Consider the map H ilb3t+1(Σ)→G(4,14)defined by C→<C>.By Lemma 2.5,the subset C(S)is simply the pullback under this map of the closed variety of spaces that intersect P9in codimension one.Therefore(a)follows,while(b)follows since no component of C is contained in S.L emma2.9.The intersection mapσ:C(S)→H ilb3S is surjective.Proof.By Corollary2.8the imageσ(C(S))is a closed subscheme of Hilb3S.Since Hilb3S is irreducible,to prove thatσis surjective it is enough to see thatσis dominant.For a point x∈S denote by P2x⊂P5=P(V)the Lagrangian plane of x,and let U={ξ∈H ilb3S:ξ=x+y+z is reduced and such thatthe Lagrangian planes P2x,P2y and P2z are mutually disjoint}. Clearly U⊂H ilb3S is open and dense,so it rests to see thatFor anyξ=x+y+z∈U there exists a smooth twisted cubic C⊂Σthat passes through x,y and z.Let U o,U∞and U y be the Lagrangian3-spaces of x,y and z in the6-space V,i.e. P2x=P(U0),P2z=P(U∞)and P2y=P(U).Since P2x and P2z do not intersect each other,we may write:V=U0⊕U∞.Furthermore,since U0and U∞are Lagrangian,we may choose coordinates(e i,x i)on U0and(e3+i,x3+i)on U∞,i=1,2,3such that in these coordinates the formαcan be written asα=x1∧x4+x2∧x5+x3∧x6.Since P2y=P(U)is also Lagrangian,and since P2y does not intersect P2z,there exists a symmetric non-singular3×3matrix B such that the Pl¨u cker coordinates of y in the system(e i,x i),i=1,...,6are uniquely written in the formy=exp(B)=(1:B:∧2B:det(B)).For a parameter t∈C,the matrix tB correspond to a Lagrangian plane that does not intersect P2z.ThusC=C x,y,z={exp(t)=(1:tB:t2∧2B:t3det(B)),t∈C∪∞}is a smooth twisted cubic onΣthrough x=exp(0),y=exp(1)and z=exp(∞).P roposition2.10.Let S=P9∩Σbe a smooth linear section with no rational curves of degree less than four,and let C(S)be the Hilbert scheme of twisted cubic curves on Σthat intersect S in a scheme of length three.Then the restriction mapσ:C(S)→H ilb3S C→C∩Sis an isomorphism.In particular C(S)is a smooth projective variety.Proof.First we show thatσis bijective.Letξ∈H ilb3(S)let P be the span ofξand assume that C1,C2∈C(S)with C i∩S=ξfor i=1,2.Let q∈P be a general point.Since S contains no plane curves,P is not contained inΩ,so we may assume that q does not lie inΩ.Therefore there is a unique line through p that intersectsΣin a scheme of length2.On the other hand for each i, there is a unique line through q that meets C i in a scheme of length2.Hence these lines must coincide,and lie in P.But the only way the general point q in P can lie on a unique line throughξ,that intersectsξin a scheme of length2,is thatξis thefirst order neighborhood of a point,i.e.when P is the tangent plane to S at the support p ofξ.In this case both C1and C2must be singular at p and have at least one line component.In fact,if the tangent cone to C i is planar,then this plane must contain the line component through p.Since S does not contain lines,the tangent cone must span the3-space,and each C i is contained in the tangent cone toΣat p.But this tangent cone is the cone over a Veronese surface(cf.[8]).In the projection from p, the plane P is mapped to a line L in the space of the Veronese surface.The line L does not intersect the Veronese surface,so it is not contained in the cubic hypersurface secant variety of the Veronese surface.It is a well known classical fact that there is a unique plane through L that meet V in a scheme of length3.This plane and the plane P spans a P3that intersectsΣalong three lines through p.Therefore,also in this case C1and C2must coincide.Next,we consider ramification of the mapσ.Consider the morphism g:C(S)→G(4,14),defined by[C]→[<C>],and similarly g3:H ilb3S→G(3,10)defined by [ξ]→[<ξ>].By Lemma2.5,both g and g3are embeddings.Therefore the restriction mapσfactors through g,the restriction map[P3]→[P3∩P9]and the inverse of g−13 restricted to Im g3.Hence a point of ramification forσis also a point of ramification for the restriction map[<C>]=[P3]→[P3∩P9]=[<ξ>].If[<C>]is such a point,then there is a tangent line to Im g at the point[<C>]∈G(4,14)that is collapsed by the restriction map.But this occurs only if the tangent line is contained in G(4,14)and parameterizes P3s through<ξ>in a P4.Since<C>∩Σ=C, andσis a bijective morphism,no other P3in this pencil can intersectΣin a twisted cubic curve.On the other hand,the ramification means that the doubling of<C> in P4intersectΣin a doubling D of C,i.e.D is a nonreduced curve of degree6.For each point p∈C consider the span P p of the tangent cone to D at p.On the one hand,any line in P p through p is tangent to D.On the other hand,the dimension of P p is one more than the dimension of the span of the tangent cone to C at p.Notice that if p and q are distinct points on C,and the linepq=∅;otherwisepr andΩ.As p and q moves,the linesT heorem3.1.Let S=P9∩LG(3,V)⊂P(W)be a smooth linear section with no rational curves of degree less than four,then H ilb3S admits afibration f:H ilb3S→P3S.(a)For any h∈P3S thefiber f−1(h)=A h=σ(C(X h)where C(X h)is the Hilbert scheme of twisted cubic curves on the threefold X h=Σ∩P10h⊃S andσis the isomorphism above,defined by C→C∩P9.(b)For the general h∈P3S thefiber A h of f is an abelian3-fold which is a La-grangian submanifold of the hyperk¨a hler6-fold H ilb3S,i.e.f:H ilb3S→P3Sis a Lagrangian abelianfibrationx in V C the hyperplane L(x)intersects V C in the plane P x(through x)and a quadric surface Q x=P1×P1.So for each point in Q x there is a unique line in Q x that meet every plane in V C in a point.Consider the three points x i and the corresponding quadrics Q i=P1×P1.Since x i∈P,the hyperplane L(x i)contains P,so x j and x k also lies in L(x i).Therefore x j,x k∈Q i,and L j,L k lies in Q i.Let x∈L(x j)be a general point,then L(x)by symmetry contains x i,so L i must lie in the quadric Q x. In particular the hyperplane L(x)contains L i.Therefore the three lines L1,L2,L3are conjugate with respect toα.The corresponding Segre threefold inΣcontains C and the point of P,so by[IR]Lemma3.2.5it is contained in H t.Recall from[8]the following propositionP roposition3.4.For each t∈F∗\Ω∗⊂P(W∗),the tangent hyperplane section H t⊂Σprojects from its point v(t)of tangency to a linear section of G(2,6),hence it admits a rank2vector bundle E t.Every Segre threefolds inΣthat pass through v(t)is the zero locus of a section of this vector bundle,and the general section of E t is of this kind.R emark3.5.For the general K3-surface section S ofΣ,the set of tangent hyperplane sections H t that contain S form the Sp(3)-dual quartic surface T=F S,hence the same surface T parameterizes rank2vector bundles E on S with determinant H and H0(S,E)=6.The general section of a vector bundle E vanishes along a subscheme of length6on S.R emark3.6.Similarly for a Fano threefold section X ofΣ,the set of tangent hy-perplane sections H t that contain X form the Sp(3)-dual quartic curve F X,hence the same curve F X parameterizes rank2vector bundles E on X with determinant H and H0(X,E)=6.The general section of a vector bundle E vanishes along an elliptic curve of degree6,a codimension2linear section of a Segre threefold.L emma3.7.Consider a Fano threefold section X=P10∩Σ⊂P(W).Let C be a twisted cubic curve contained in X,and let H t be a singular hyperplane section ofΣthat contains X.Then there is a unique twisted cubic curve C′⊂X with length(C∩C′)=2 determined by t.In particular,X admits a rank2vector bundle E t and a unique section s∈Γ(X,E t),that Z(s)=C∪C′.Proof.Let Y C,t be the unique Segre3-fold through C that is contained in the tangent hyperplane section H t ofΣ.Then Y C,t∩X=C∪C′is a codimension2linear section of a Segre3-fold.In particular C′is a twisted cubic curve and length(C∩C′)=2. Thefinal statement follows from Proposition3.4and Remark3.6.Consider the incidenceI S,T={(ξ,t)∈H ilb3S×T|H0(S,E t⊗Iξ)>0}Clearly,by the aboveI S,T={(ξ,t)|Cξ⊂H t}where Cξis the unique twisted cubic curve throughξ.Now,T=F S is the Sp(3)-dual quartic surface to S,henceTξ={t∈T|(ξ,t)∈I S,T}is the plane quartic curve on T which is Sp(3)-dual to the threefold<S∪Cξ>∩Σ. Thereforeξ→Tξdefines again the mapf:Hilb3S→|Tξ|=P3S.Let Aξbe thefiber f−1(f(ξ)).Letη∈Aξbe a general point in thefiber.By Lemma3.7there is a morphismτη:Tξ→Aξwhich assigns to a point t∈Tξthe element(C′∩S)∈Aξ,where C′is the unique twisted cubic curve residual to Cηin the Segre3-fold defined by Cηand t.By uniqueness, this map is injective:The curve Cηis the unique twisted cubic curve residual to C′determined by t.ThusP roposition3.8.Let S=P9∩Σbe a smooth linear section with no rational curves of degree less than four.Let T=F S be the Sp(3)-dual quartic surface,and consider the abelianfibration f:Hilb3S→|Tξ|=P3S,where Tξis the plane section of T determined by Cξ,the unique twisted cubic curve inΣthroughξ.Let Aξbe thefiber f−1(f(ξ)). When Tξis smooth,then any elementη∈Aξdefines an embeddingτη:Tξ→Aξ.4Twisted cubic curves on the prime Fano three-folds of genus9In the previous section we concluded that thefibers of thefibrationf:Hilb3S→P3Sare abelian threefolds.From the construction we note that thefiber is identified with the subset of C(S)of twisted cubic curves C that are contained in afixed P10h⊃S.In particular,the generalfiber coincides with the Hilbert scheme of twisted cubic curves in the Fano threefold X h=Σ∩P10h.The Sp(3)-dual variety to X h is a plane quarticcurve F Xh ,the plane section T h=T∩P2h of the Sp(3)-dual surface T to S,whereP2h=(P10h)⊥⊂P(W∗).In this section we shall prove the following theorem:T heorem4.1.Let S=P9∩Σ⊂P(W)be a smooth linear section with no rational curves of degree less than four.Let T=F S be the Sp(3)-dual quartic surface,and consider the abelianfibration f:Hilb3S→|T h|=P3S.The generalfiber A h=f−1(h) is isomorphic to the jacobian of the Sp(3)-dual plane quartic curve T h to the Fano 3-fold X h⊃S.For the proof we shall need to know some additional properties of the Hilbert scheme C(X)of twisted cubic curves on the general prime Fano threefold X of genus9.We begin with:P roposition4.2.Let X=P10∩Σbe a general prime Fano threefold of index1 and genus9,and let F X be its Sp(3)-dual plane quartic curve.Then the intermediate jacobian of X is isomorphic to the jacobian J(F X)of F X.Proof.In[13],Mukai identifies the intermediate jacobian of X with the Jacobian of a curve of genus3,and in[7]this curve is identified with F X.We shall show that the family C(X)is isomorphic to the intermediate jacobian J(X).For this we shall use the birational properties of the Fano3-fold X of genus9 related to twisted cubic curves on X.Let B⊂X⊂P10be a smooth rational normal cubic curve,and consider the rational projectionπ:X···>P6from the space P3=<B>.Let X′⊂P6be the properπ-image of X,and let β:˜X→X be the blowup of X at B.Since we may assume that<B>∩X=B, the blowupβresolves the indeterminacy locus ofπ,so the projectionπextends to a morphism˜π:˜X→X′.Recall from§4.1in[6]that the cubic B⊂X fulfills the conditions(*)-(**)if:(*).the anticanonical divisor−K˜X is numerically effective(nef)and(−K˜X)3>0(i.e.−K˜X is big);(**)there are no effective divisors D on X such that(−K˜X)2.D=0.By the Remark on page66in[6],the twisted cubic B⊂X will satisfy the conditions (*)-(**)if the morphism˜πhas only afinite number offibers of positive dimension.L emma4.3.Suppose that X=X16is general and the smooth twisted cubic B⊂X is general.Then the morphism˜πhas only afinite number of irreduciblefibers of positive dimension,and these are precisely the properβ-preimages of the e=e(B)=12lines on X that intersect the twisted cubic curve B.Proof.First,we may assume that<B>∩X=B by Lemma2.5and that B is irreducible.The family of planes in P5parameterized by B sweeps out an irreducible 3-fold of degree3.If this3-fold is singular,it is a cone,and B⊂Q p for some p∈P5, contrary to the assumption.Therefore this3-fold is smooth,isomorphic to the smooth Segre embedding of P1×P2.Thus any two planes representing points of B are disjoint; in particular no conic in X intersects B in a scheme of length2,since the Lagrangian planes representing the points of a conic in X always have a common point.。