DOCSLIB.ORG

Some Recent Contributions to Algebraic Geometry*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Some Recent Contributions to Algebraic Geometry* SOME RECENT CONTRIBUTIONS TO ALGEBRAIC GEOMETRY* BY VIRGIL SNYDER 1. Introduction. During the last decade mathematical litera­ ture has been enriched by over a thousand contributions to algebraic geometry, including about eight hundred to the nar­ rower field of rational transformations. It would be an ambitious task to report on this immense field in one short talk. Today I wish to speak of only three problems, each one somewhat well defined, or even narrow. Of these, one furnishes a striking ex­ ample of mathematical elegance in providing one solution to what was regarded as several distinct problems. The other two are capable of unlimited extension, each new enlarged field furnishing phases not existing in the earlier ones. 2. Series of Composition of Veneroni Transformations. The Cremona transformations determined by a system of bilinear equations between the systems of coordinates of the two associ­ ated spaces were among the first to be considered in 52 and 53. For the general case the configuration of fundamental and principal elements can be at once expressed in terms of the vanishing of certain determinants of a matrix. (Segre.f) This has recently been generalized to spaces of higher dimensions by various authors, in particular to Sn by Godeaux.J Numerous particular cases have been considered. If I, the ex­ treme case of inversion results. Transformations made up of this inversion and of collineations have no fundamental curves of the first kind ; they have a series of composition somewhat similar to that of the general case for 52. If in every bilinear equation the coefficients a»* and au are equal, the transformation is involutorial and can be expressed in terms of polarity as to a series of quadrics or as to null systems. In case all the polarities are quadric and the quadric primais are independent, there are only a finite number of invariant * Presented to the Society at the symposium held in New York, March 30, 1934, at the invitation of the program committee. t C. Segre, Rendiconti dei Lincei, (2), vol. 9 (1900), pp. 253-260. ÎL. Godeaux, Lombardo Istituto Rendiconti, (2), vol. 43 (1910), pp. 116-119. 673 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 674 VIRGIL SNYDER [October, points. These have been discussed by M. del Re.* The case of w = 4 has been investigated by Miss Alderton,f and various further properties have been given by Wong{ and Roth.§ The general Cremona transformation for Sry r>2, does not have a series of composition, that is, it cannot be expressed as the product of transformations of lower order. A Cremona transformation in 53 of order n = 4& + r having as base curve of maximal multiplicity a sextic curve C of genus p — Z and of multiplicity & + 1, is the product of a (3, 3) and of another transformation of order n. (Piazzolla-Beloch.||) Similarly, a T^n-i'- Cz{n~l\ the only other base-system con­ sisting of bisecants of C3, has a definite series of composition. (Montesano.lf) This case has been developed further by Tinto.** 3. Case n = 3. The most important particular case from this point of view for n = 3 is that in which the base C% is composed of four skew lines r» and their two transversals u} v. The inverse is defined by cubic surfaces through four skew lines ri and their two transversals u', v'. A point on /%• has for image a line meet­ ing three lines ri, so that r{ itself is transformed into the quadric through three lines ri . A point on u (or v) has the whole line u' (or v') for image, and conversely. These transversals are simple fundamental lines of the second kind. The image of an arbitrary line is a space cubic meeting each ri twice, but not meeting uf or v'. A line r meeting u and v is transformed into a line meeting uf and v'. The (1, 1) corre­ spondence between this line and its image is a collineation, such * Maria del Re, Naples Rendiconti, (3), vol. 28 (1922), pp. 203-211, and (3), vol. 35 (1929), pp. 208-217. f Nina Alderton, University of California Publications in Mathematics, vol. 1 (1923), pp. 345-358. t B. C. Wong, Annals of Mathematics, (2), vol. 27 (1926), pp. 330-336, and vol. 28 (1927), pp. 251-262; American Journal of Mathematics, vol. 49 (1927), pp. 383-388; this Bulletin, vol. 35 (1929), pp. 829-832. § L. Roth, Proceedings of the Cambridge Philosophical Society, vol. 29 (1933), pp. 178-183. || M. Piazzolla-Beloch, Annali di Matematica, (3), vol. 16 (1909), pp. 27-98. If D. Montesano, Naples Rendiconti, (3), vol. 27 (1921), pp. 116-127 and 164-175. ** F. J. Tinto, Proceedings of the Edinburgh Mathematical Society, vol. 34 (1916), pp. 133-145. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use I934-] RECENT ALGEBRAIC GEOMETRY 675 that the points r, u and r, v are transformed into r', u' and r', v'. Through a point P in (x) passes a single line meeting w, A; through its image P' passes a single line meeting u', v'. The col- lineation is now uniquely fixed. Let u be taken as Xi = 0, #2 = 0 and v as x3 = 0, #4 = 0. For fi, r2 the lines #i = 0, x3 = 0 and x2 = 0, : #4 = 0 may be taken. Then for r3 take axi + bx2= 0J cxs+dxi~0 and r4' is a'xi + b'x2 = 0, c'x3+d'x4 = 0. Since the quadric surface through any three lines rt- contains the two transversals, a quadric and any plane of the pencil through the residual line will be a cubic surface of the homa- loidal system. If Hn = 0 is the equation of the quadric through all the lines except X\ — 0, x3 = 0, etc., we may write X X\ = XiHu, %2 = X2#24, i — #3#13, Xl = XAH2±. Similar pencils are of the forms (a%i + bx2jHa,c + (£#3 + dxi)Ha,c = 0, (arxi + bfX2)Ha',c' + etc. = 0. By writing the equations of Hn, H2A we can now obtain the forms of the defining bilinear equations Xi X3 — X\Xz = 0, X{ #4 — X2x{ = 0, (cd')(axi + bx2)(b'xi — a'xl) — {ab'){d'x( — c'x2)(cxz + dx±) = 0. 4. Products of (3, 3) Cremona Transformations. Now consider f the product of TiifiU, v; r'u , v' and T2\rlu', v'\ r{ , w", z/' in which the same lines #', */ appear in Trl and in TV The lines ri, r/ may all be distinct or in part the same. Similarly for the continued product Ti, T2, • • • , 7V For every value of k the transformation is determined by two lines u, v of the same multi­ plicity and of a number of lines, of various multiplicities, meeting both of them, and of no other base elements. Conversely, consider a transformation T defined by the two properties : (a) To a line r meeting two skew lines u, v shall correspond a line r' meeting two skew lines u', v'. (b) The projectivity established between r and r' shall never be degenerate, and the point r, u shall correspond to the point r', u'\ similarly for r9 v and r\ v'. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 676 VIRGIL SNYDER [October, The transformations determined by these conditions have u, v to the same multiplicity v\ a number of lines meeting both are base elements of the web, and there can be no other base ele­ ments. The transformation is of order 2^ + 1. We may then write Ici I JL JL I • V V Pl PP | 02 I ~ I <j>2v </>2v+l \ .UVTi ' ' ' rp , Ici \n n \ . 2pi 2Pp I ài J ~ | C2^C2^+i J .r-i - • • rp . A plane through u (or v) is transformed into a ruled surface v v 1 2 2p R2v+i:u' v' + r Pi • • • rp *. Each line r» is transformed into a ruled surface; each point of Yi~a. rational curve o' of order p/, not meeting #', z>', which ,pl p generates R2pi\u ' v' >'. An arbitrary plane co meets w, z; each in a point and contains the line / joining them. Let co' be any plane (not containing u', v') through the image of this line. Given any point T in co, through it passes a unique line c meeting u, v. The image line c' meets u', v' and pierces co' in T'. Between co, co' exists a plane Cremona transformation. Let T describe a line p in co. The lines u, v, p determine a quadric surface, the image of which is a sur­ face of order 4^ + 2. The section by co' contains V and the image of p} hence Since in a plane Cremona transformation the number of F points in the two planes is the same, it follows that p — p'. From the two fundamental relations of a plane Cremona transformation, we have further JJD*^, ^P*2 = 2v(v + l). Let Pi^P2^ • • • 'èPp. Then 2 2 2 Pi + P2 + P3 + p4(p4 + • • • + Pp) è 2v(v + 1), Pi2 + P22 + P32 + PA(4V - PI - P2 - Pa) è 2v(v + 1), v(pi — p4) + y(p2 — p4) + ^(P3 ~ P4) + 4ï>P4 è 2p(l' + 1), Pl + P2 + P3 + P4 à 2(v + 1) . From this last equation it follows that if the web of surfaces, homaloids of the planes of space, is transformed by means of the (3, 3) transformation defined by ru r2, n$, r^ the order of the surfaces of the web will be lowered.
Load more

Recommended publications
  • Fano Hypersurfaces with Arbitrarily Large Degrees of Irrationality 3
    FANO HYPERSURFACES WITH ARBITRARILY LARGE DEGREES OF IRRATIONALITY NATHAN CHEN AND DAVID STAPLETON Introduction There has been a great deal of interest in studying questions of rationality of various flavors. Recall that an n-dimensional variety X is rational if it is birational to Pn and ruled if it is birational to P1 Z for some variety Z . Let X Pn+1 be a degree d hypersurface. × ⊂ In [8], Kollár proved that when X is very general and d 2 3 n + 3 then X is not ruled ≥ ( / )( ) (and thus not rational). Recently these results were generalized and improved by Totaro [14] and subsequently by Schreieder [13]. Schreieder showed that when X is very general and d log n + 2 then X is not even stably rational. In a positive direction, Beheshti and Riedl ≥ 2( ) [2] proved that when X is smooth and n 2d! then X is at least unirational, i.e., dominated ≥ by a rational variety. Given a variety X whose non-rationality is known, one can ask if there is a way to measure “how irrational" X is. One natural invariant in this direction is the degree of irrationality, defined as irr X ≔ min δ > 0 ∃ degree δ rational dominant map X d Pn . ( ) { | } For instance, Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery [1] computed the degree of irrationality for very general hypersurfaces of degree d 2n + 1 by using the positivity of the ≥ canonical bundle. Naturally, one is tempted to ask what can be proved about hypersurfaces with negative canonical bundle. Let X Pn+1 be a smooth hypersurface of degree d.
    [Show full text]
  • Resonance Varieties Over Fields of Positive Characteristic
    Resonance varieties over fields of positive characteristic Michael J. Falk July 11, 2018 Abstract Let A be a hyperplane arrangement, and k a field of arbitrary char- acteristic. We show that the projective degree-one resonance variety R1(A, k) of A over k is ruled by lines, and identify the underlying algebraic line complex L(A, k) in the Grassmannian G(2, kn), n = |A|. L(A, k) is a union of linear line complexes corresponding to the neighborly partitions of subarrangements of A. Each linear line complex is the intersection of a family of special Schubert varieties corresponding to a subspace arrange- ment determined by the partition. In case k has characteristic zero, the resulting ruled varieties are lin- ear and pairwise disjoint, by results of A. Libgober and S. Yuzvinsky. We give examples to show that each of these properties fails in positive char- acteristic. The (4,3)-net structure on the Hessian arrangement gives rise 1 4 to a nonlinear component in R (A, Z3), a cubic hypersurface in P with interesting line structure. This provides a negative answer to a question of A. Suciu. The deleted B3 arrangement has linear resonance components over Z2 that intersect nontrivially. 1 Resonance and characteristic varieties Arising out of the study of local system cohomology and fundamental groups, arXiv:math/0611008v1 [math.CO] 1 Nov 2006 characteristic and resonance varieties of complex hyperplane arrangements have become the object of much of the current research in the field. The study of resonance varieties in particular has led to surprising connections with other areas of mathematics: generalized Cartan matrices [23], Latin squares and loops, nets [40], special pencils of plane curves [23, 13], graded resolutions and the BGG correspondence [10, 33, 34], and the Bethe Ansatz [7].
    [Show full text]
  • Title Deformations of Irrational Affine Ruled Surfaces Author(S)
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Kyoto University Research Information Repository Title Deformations of irrational affine ruled surfaces Author(s) Kishimoto, Takashi Citation 代数幾何学シンポジウム記録 (2013), 2013: 25-32 Issue Date 2013 URL http://hdl.handle.net/2433/215002 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University 代数幾何学シンポジウム記録 2013年度 pp.25-32 DEFORMATIONS OF IRRATIONAL AFFINE RULED SURFACES TAKASHI KISHIMOTO Abstract. This note consists in a report of my talk about deformations of affine surfaces possessing A1-fibrations at Kinosaki conference, which concerns a joint work with Adrien Dubouloz (L'Institut de Math´ematiques,l'Universit´ede Bourgogne) [DuKi13]. Certainly the structure of affine surfaces admitting A1-fibrations themselves are well understood nowadays, but that of deformations of such surfaces is quite complicated to grasp. We mention a factorization theorem in case of a deformation of irrational affine surfaces with A1-fibrations by noting that such a factorization is not feasible in general in case of a deformation of rational ones. 1. Introduction & Motivation 1.1. All varieties treated in this article are defined over the field of complex numbers C. As a general perception, projective varieties covered by images of projective line P1, so-called uniruled varieties, play important roles in the theory of birational geometry as outputs of minimal model program. As an affine counterpart of this concept, affine algebraic varieties swept out by images of the affine line A1, so-called affine uniruled varieties, should play still important roles in order to deal with problems on polynomial rings by means of geometry.
    [Show full text]
  • Lecture Notes in Mathematics 1842 Editors: J.--M.Morel, Cachan F
    Lecture Notes in Mathematics 1842 Editors: J.--M.Morel, Cachan F. Takens, Groningen B. Teissier, Paris 3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo Trygve Johnsen Andreas Leopold Knutsen K3 Projective Models in Scrolls 13 Authors Trygve Johnsen Department of Mathematics University of Bergen Johs. Bruns gt. 12 5008 Bergen, Norway e-mail: [email protected] Andreas Leopold Knutsen Department of Mathematics University of Oslo Box 1053, Blindern 0316 Oslo, Norway e-mail: [email protected] LibraryofCongressControlNumber:2004103750 Mathematics Subject Classification (2000): 14J28, 14H51 ISSN 0075-8434 ISBN 3-540-21505-0 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag is a part of Springer Science + Business Media http://www.springeronline.com c Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEXoutputbytheauthor SPIN: 10999523 41/3142/ du - 543210 - Printed on acid-free paper Preface The cover picture shows a smooth quartic surface in space, the simplest ex- ample of a projective model of a K3 surface.
    [Show full text]
  • Algebraic Geometry and Local Differential Geometry
    ANNALES SCIENTIFIQUES DE L’É.N.S. PHILLIP GRIFFITHS JOSEPH HARRIS Algebraic geometry and local differential geometry Annales scientifiques de l’É.N.S. 4e série, tome 12, no 3 (1979), p. 355-452. <http://www.numdam.org/item?id=ASENS_1979_4_12_3_355_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1979, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systéma- tique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. scient. EC. Norm. Sup. 46 serie, t. 12, 1979, p. 355 a 432. ALGEBRAIC GEOMETRY AND LOCAL DIFFERENTIAL GEOMETRY BY PHILLIP GRIFFITHS (1) AND JOSEPH HARRIS (1) CONTENTS Introduction. ................................................................. 356 1. Differential-geometric preliminaries ................................................ 360 (a) Structure equations for the frame manifolds. ........................................ 360 (b) The 2nd fundamental form. .................................................... 363 (c) Examples ................................................................. 368 (d) The higher fundamental forms .................................................. 372 (e)
    [Show full text]
  • RESEARCH STATEMENT Part 1. Research Summary 1. Every
    RESEARCH STATEMENT JASON STARR Part 1. Research summary 1. Every rationally connected variety over the function field of a curve has a rational point A nonempty, smooth, projective variety over an uncountable, algebraically closed field is rationally connected if every pair of closed points is in the image of a regular morphism from P1 to the variety. A smooth, projective variety over an arbitrary field k is rationally connected if its base-change to one (and hence every) uncount- able, algebraically closed field is rationally connected. Theorem 1.1 (Graber, Harris, Starr [12]). Every rationally connected variety X defined over the function field K of a curve B over a characteristic 0, algebraically closed field k has a K-rational point. Equivalently, every projective, surjective mor- phism π : X! B whose general fiber is rationally connected has an algebraic section. Theorem 1.2 (de Jong, Starr [4]). Replacing \rationally connected" by \separably rationally connected", the previous theorem holds in arbitrary characteristic. Theorem 1.1 was posed as a question by J. Koll´ar,Y. Miyaoka and S. Mori in their paper [21], and proved by them when dim(X) is 1 or 2. Also, Also, the special case of a smooth Fano hypersurface was proved by Tsen [19, Thm IV.5.4]. Corollary 1.3. Let (R; m) be a complete DVR containing its residue field, assumed algebraically closed. Let X be a regular, projective R-scheme. If the geometric generic fiber X ⊗R K(R) is normal and separably rationally connected, then the closed fiber X ⊗R k is reduced on a nonempty open subset.
    [Show full text]
  • Rational and Non-Rational Algebraic Varieties: Lectures of J\'Anos Koll\'Ar
    RATIONAL AND NON-RATIONAL ALGEBRAIC VARIETIES: LECTURES OF JANOS´ KOLLAR´ By Karen E. Smith with an Appendix by Joel Rosenberg July 14, 1997 Introduction Rational varieties are among the simplest possible algebraic varieties. Their study is as old as algebraic geometry itself, yet it remains a remarkably difficult area of research today. These notes offer an introduction to the study of rational varieties. We begin with the beautiful classical geometric approach to finding ex- amples of rational varieties, and end with some subtle algebraic arguments that have recently established non-rationality of varieties that otherwise share many of their traits. In lecture one, rationality and unirationality are defined, and illustrated with a series of examples. We also introduce some easily computable invariants, the pluri- genera, that vanish for all rational varieties. Using the plurigenera, one develops a sense how rare rational varieties are. For example, we will immediately see that no smooth projective hypersurface whose degree exceeds its embedding dimension can be rational: no plane cubic curve is rational, no space quartic surface is rational, and so on. arXiv:alg-geom/9707013v1 15 Jul 1997 The second and third lectures focus on the rationality question for smooth cubic surfaces over arbitrary fields, an issue thoroughly explored by B. Segre in the forties. This is a pretty story, depending subtly on the field of definition. It was already understood one hundred years ago that every cubic surface is rational over the complex numbers; however, the situation is quite complicated over the rational numbers. In the second lecture, we construct examples of non-rational smooth cubics over Q by considering the orbits of the Galois group of Q¯ /Q on the twenty seven lines on the cubic surface.
    [Show full text]
  • Measuring the Irrationality of Abelian Surfaces and Complete Intersections
    Measuring the Irrationality of Abelian Surfaces and Complete Intersections A Dissertation Presented by Nathan Chen to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics Stony Brook University May 2021 Copyright by Nathan Chen 2021 Stony Brook University The Graduate School Nathan Chen We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation Robert Lazarsfeld Distinguished Professor of Mathematics Samuel Grushevsky Professor of Mathematics Olivier Martin James H. Simons Instructor, Department of Mathematics Lawrence Ein LAS Distinguished Professor of Mathematics University of Illinois at Chicago This dissertation is accepted by the Graduate School Eric Wertheimer Dean of the Graduate School ii Abstract of the Dissertation Measuring the Irrationality of Abelian Surfaces and Complete Intersections by Nathan Chen Doctor of Philosophy in Mathematics Stony Brook University 2021 In this dissertation, we make advances in the study of measures of irrationality for polar- ized abelian surfaces and codimension two complete intersections, which answer a number of questions posed in [7]. iii Table of Contents Acknowledgementsv Introduction1 1 Measures of irrationality: background and survey2 1.1 Measures of irrationality.............................4 1.2 Beyond hypersurfaces of large degree......................7 2 Abelian surfaces 11 2.1 Examples..................................... 12 2.2 Kummer construction and even sections.................... 14 2.3 Constructing maps from sections of H0(A; 2L)+ ................ 17 2.4 Constructing elliptic fibrations with sections on Kummer K3 surfaces.... 23 3 Complete intersections 26 3.1 Covering gonality and separating points..................... 28 3.2 Reduction step.................................. 30 3.3 Multiplicative bounds for complete intersection curves............
    [Show full text]
  • Rational Curves on Algebraic Varieties
    Rational curves on algebraic varieties ||||||{ Lecture notes for the GAeL XVIII conference Coimbra, Portugal { June 6-11, 2010 and for the Semaine sp´ecialeMaster de math´ematiques Universit´ede Strasbourg { May 2-6, 2011 ||||||{ Olivier Debarre August 29, 2011 Contents 1 Rational curves 3 1.1 Exceptional locus of a morphism . 3 1.2 Rational curves and birational morphisms . 3 1.3 Parametrizing rational curves . 5 1.4 Parametrizing morphisms . 6 1.5 Parametrizing morphisms with extra structure . 9 1.6 Lines on a subvariety of a projective space . 11 1.7 Bend-and-break . 13 1.8 Rational curves on varieties whose canonical divisor is not nef . 16 2 Varieties with many rational curves 20 2.1 Rational varieties . 20 2.2 Unirational and separably unirational varieties . 20 2.3 Uniruled and separably uniruled varieties . 22 2.4 Free rational curves and separably uniruled varieties . 23 2.5 Rationally connected and separably rationally connected varieties 28 2.6 Very free rational curves and separably rationally connected va- rieties . 30 2.7 Smoothing trees of rational curves . 34 2.7.1 Trees . 34 2.7.2 Smoothing trees with free components . 36 2.7.3 Smoothing combs . 36 2.7.4 Another smoothing result for combs . 40 1 2.8 Rationally chain connected varieties . 42 3 Sections of families of separably rationally connected varieties 46 3.1 (C1) fields . 46 3.2 Sections of families of separably rationally connected varieties . 47 3.2.1 Strategy of the proof of the theorem . 49 3.2.2 Case where all fibers are reduced .
    [Show full text]
  • The Stability of Certain Classes of Uni-Ruled Varieties Compositio Mathematica, Tome 52, No 1 (1984), P
    COMPOSITIO MATHEMATICA MARC LEVINE The stability of certain classes of uni-ruled varieties Compositio Mathematica, tome 52, no 1 (1984), p. 3-29 <http://www.numdam.org/item?id=CM_1984__52_1_3_0> © Foundation Compositio Mathematica, 1984, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 52 (1984) 3-29 © 1984 Martinus Nijhoff Publishers, The Hague. Printed in The Netherlands THE STABILITY OF CERTAIN CLASSES OF UNI-RULED VARIETIES Marc Levine Introduction The first results on the deformations of ruled varieties were obtained by Kodaira and Spencer [8], where they show that all small deformations of a ruled surface are also ruled. Their argument uses the Castelnuovo-En- riques criterion for ruledness in an essential way, and thus, in view of the absence of a similar criterion for varieties of higher dimension, their method cannot be applied to the study of the deformations of a general ruled variety. In fact, our paper [9] shows that the straightforward generalization of the Kodaira-Spencer theorem is false, in that there is a ruled threefold that can be deformed into a threefold that is not ruled. Here we will study the deformations of ruled varieties, and other related varieties, and we will give some natural conditions which allow one to recover the stability results.
    [Show full text]
  • Arxiv:Math/0401027V5
    ON HIGHER SYZYGIES OF RULED VARIETIES OVER A CURVE EUISUNG PARK Abstract. For a vector bundle E of rank n + 1 over a smooth projective curve C of genus g, let X = PC (E) with projection map π : X → C. In this paper we investigate the minimal free resolution of homogeneous coordinate rings of X. We first clarify the relations between higher syzygies of very ample line bundles on X and higher syzygies of Veronese embedding of fibres of π by the same line bundle. More precisely, letting H = OPC (E)(1) be the tautological line bundle, n ∗ we prove that if (P , OPn (a)) satisfies Property Np, then (X,aH + π B) satisfies Property Np for all B ∈ PicC having sufficiently large degree(Theorem 1.1). And also the effective bound of deg(B) for Property Np is obtained(Theorem 4.2, 4.5, 4.7 and 4.9). For the converse, we get some partial answer(Corollary 3.3). Secondly, by using these results we prove some Mukai-type statements. In particular, Mukai’s conjecture is true for X when rank(E) ≥ g and µ−(E) is an integer(Corollary 4.11). Finally for all n, we construct an n-dimensional ruled variety X and an ample line bundle A ∈ PicX which shows that the condition of Mukai’s conjecture is optimal for every p ≥ 0. Contents 1. Introduction 1 2. Notation and Preliminaries 4 3. Veronese Embedding and Property Np of Ruled Varieties 7 4. Property Np of Ruled Varieties over a Curve 11 5. Optimality of the condition of Mukai’s conjecture 21 References 23 1.
    [Show full text]
  • Projective Normality of Special Scrolls
    Projective normality of special scrolls. Luis Fuentes Garc´ıa∗ Manuel Pedreira P´erez Authors’ address: Departamento de Algebra, Universidad de Santiago de Compostela. 15706 Santiago de Compostela. Galicia. Spain. e-mail: [email protected]; [email protected] Abstract: We study the projective normality of a linearly normal special scroll R of degree d and speciality i over a smooth curve X of genus g. If d ≥ 4g − 2i + 1, we prove that the projective normality of the scroll is equivalent to the projective normality of its directrix curve of minimum degree. Mathematics Subject Classifications (1991): Primary, 14J26; secondary, 14H25, 14H45. Key Words: Ruled Surfaces, projective normality. Introduction. Let R ⊂ PN be a scroll, that is, a surface with a one-dimensional family of lines. We say that R is linearly normal when R is not the projection of any other scroll with the same degree and genus. It is known that there are a unique geometrically ruled surface π : S = P(E) −→ X and a unique base point free unisecant linear system |H| on S such that R is the birational image of S = P(E) by the regular map defined by the complete linear system |H|. We 1 define the speciality of R as i = h (OS (H)). The scroll is special when i> 0. In this paper we study the projective normality of a special scroll R. We say that R is projectively normal, or equivalently, that the line bundle OS(H) on S 0 is normally generated, if and only if the natural maps Symk(H (OS(H))) −→ 0 H (OS (kH)) are surjective for any k ≥ 1.
    [Show full text]