低维射影簇的构造性研究
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摘要
构造性方法是有效的解决代数几何问题的方法之一。本文用构造性方法研究一系列的低维射影簇一即奇异平面六次代数曲线,和含有很多直线的五次曲面。第一部分研究的是平面六次曲线的Zariski pair,我们利用Urabe的格的方法,Yang的算法以及Shimada的不变量来寻找到很大一类利用格区分的Zariskipair及Zariski triplet,配合上Shimada的结果,我们得到结论,只有简单奇点的平面六次曲线的具有相同的Configuration,但是discriminant group的阶不同的Zariski pair共有123对,Zariski triplet共有4组。
第二部分是处理极大六次曲线的多项式的构造问题。在用格的办法确定了相应的奇异曲线的Configuration后,我们关心如何得到它的一个解析的实现。这章中,我们介绍了判别局部结构的一些基础知识,回顾了前人在构造曲线的多项式方程的结果,并把问题集中到Milnor数等于19的,只含有二重点的,不可约的六次曲线的构造上。与以前的方法不同的是,我们用的主要工具是三类Cremona变换,并且利用了一个六次曲线的pencil,用pencil中的两重的三次曲线来逼近待求的曲线,并且证明这个pencil的退化成员在Cremona变换下能分裂出若干直线成员。从而作为双有理像的pencil就由比较简单的代数方程组来表达和求解。配合上前人的工作,除了两个Configuration外,我们得到了上述的39个Configuration中的37个。在本章中,我们还从定义出发,给出了Configuration的组合的定义,从而可以定义组合的算法来进行相应的双有理变换的演算。
第三部分是计数代数几何中的一个问题,我们从Segre的有关光滑五次曲面上的直线条数的最大值的问题出发,考虑在Dwork pencil中构造一个对称的,含有很多直线的五次曲面.我们利用对称群和相应的组合的技巧,完整地刻画了这个pencil,解出了里面的5个奇异曲面(除去由五个坐标面组成的)。然后利用直线的相交形式证明如果有基点外直线,那么它必与至少两条包含在基点中的直线相交,从而唯一地决定了可能的直线的显式表达,以及所在的曲面。最后我们说明,只有三类光滑曲面含有基点以外的直线,所含有的直线条数是35,55和75条,并且它们都不与已知的Fermat五次曲面同构。
Constructive method is one of the most effective method in studying concrete projective varieties.This thesis concerns with some low dimensional projective varieties—the plane sextics with simple singularities and smooth quintic surfaces with lots of projective lines.In the first part,we consider the problem of finding Zariski pair.By the relation of K3-surface and plane sextic,we use Urabe's criterion and Yang's algorithm and Shimada's invariant to determine the Zariski pairs distinguished by lattice.With Shimada's results,we find all 123 Zariski pairs and 4 Zariski triplets whose order of the discriminant groups are different.
The second part is devoted to the construction of polynomial of the sextic with a given configuration.After the determination of the existence of a configuration by lattice method,we want to get its analytic realization.We introduce some basis on the local structure,and review some results by previous authors.After that,we concentrate to the irreducible maximal plane sextic with double points only.We employ a new method which consists of several parts.One main tool is the Cremona Transform.The other is the fact that:the target sextic sits in a pencil which contains a double cubic passing nine double points(including infinitely near ones),where the degenerated member of the pencil splits out linear members under the Cremona Transforms.Hence the virtual image of the original pencil after Cremona Transform contains a member with lots of projective lines, and then the virtual images of the cubic,the original sextic and the reducible one consist a configuration which is easier to be computed by algebraic equations. Combined with the known results,we get all the 39 such irreducible curves except two.This chapter also gives a combinatorial definition of Virtual Configuration which helps writing algorithm for applying symbolic birational transform.
In the third part,we consider a problem in the enumerative algebraic geometry. It is Segre who first partially answered the question that what is the maximal number of the lines on a smooth surface with degree equal or great than four.People refined the upper and lower bound by constructing new surfaces.We studies the Dwork pencil of quintic surfaces here.Using the some tricks and intersection form,we describes the pencil in detail.The pencil has five singular surfaces ex- cept the surface consists of the five coordinate planes.And we determine the only possible lines outside the base locus of the pencil,and the smooth surfaces holding such lines.We get the conclusion that only three classes of surfaces contain additional lines,with the number 35,55 and 75.And none of them are isomorphic to the Fermat quintic surface.
第二部分是处理极大六次曲线的多项式的构造问题。在用格的办法确定了相应的奇异曲线的Configuration后,我们关心如何得到它的一个解析的实现。这章中,我们介绍了判别局部结构的一些基础知识,回顾了前人在构造曲线的多项式方程的结果,并把问题集中到Milnor数等于19的,只含有二重点的,不可约的六次曲线的构造上。与以前的方法不同的是,我们用的主要工具是三类Cremona变换,并且利用了一个六次曲线的pencil,用pencil中的两重的三次曲线来逼近待求的曲线,并且证明这个pencil的退化成员在Cremona变换下能分裂出若干直线成员。从而作为双有理像的pencil就由比较简单的代数方程组来表达和求解。配合上前人的工作,除了两个Configuration外,我们得到了上述的39个Configuration中的37个。在本章中,我们还从定义出发,给出了Configuration的组合的定义,从而可以定义组合的算法来进行相应的双有理变换的演算。
第三部分是计数代数几何中的一个问题,我们从Segre的有关光滑五次曲面上的直线条数的最大值的问题出发,考虑在Dwork pencil中构造一个对称的,含有很多直线的五次曲面.我们利用对称群和相应的组合的技巧,完整地刻画了这个pencil,解出了里面的5个奇异曲面(除去由五个坐标面组成的)。然后利用直线的相交形式证明如果有基点外直线,那么它必与至少两条包含在基点中的直线相交,从而唯一地决定了可能的直线的显式表达,以及所在的曲面。最后我们说明,只有三类光滑曲面含有基点以外的直线,所含有的直线条数是35,55和75条,并且它们都不与已知的Fermat五次曲面同构。
Constructive method is one of the most effective method in studying concrete projective varieties.This thesis concerns with some low dimensional projective varieties—the plane sextics with simple singularities and smooth quintic surfaces with lots of projective lines.In the first part,we consider the problem of finding Zariski pair.By the relation of K3-surface and plane sextic,we use Urabe's criterion and Yang's algorithm and Shimada's invariant to determine the Zariski pairs distinguished by lattice.With Shimada's results,we find all 123 Zariski pairs and 4 Zariski triplets whose order of the discriminant groups are different.
The second part is devoted to the construction of polynomial of the sextic with a given configuration.After the determination of the existence of a configuration by lattice method,we want to get its analytic realization.We introduce some basis on the local structure,and review some results by previous authors.After that,we concentrate to the irreducible maximal plane sextic with double points only.We employ a new method which consists of several parts.One main tool is the Cremona Transform.The other is the fact that:the target sextic sits in a pencil which contains a double cubic passing nine double points(including infinitely near ones),where the degenerated member of the pencil splits out linear members under the Cremona Transforms.Hence the virtual image of the original pencil after Cremona Transform contains a member with lots of projective lines, and then the virtual images of the cubic,the original sextic and the reducible one consist a configuration which is easier to be computed by algebraic equations. Combined with the known results,we get all the 39 such irreducible curves except two.This chapter also gives a combinatorial definition of Virtual Configuration which helps writing algorithm for applying symbolic birational transform.
In the third part,we consider a problem in the enumerative algebraic geometry. It is Segre who first partially answered the question that what is the maximal number of the lines on a smooth surface with degree equal or great than four.People refined the upper and lower bound by constructing new surfaces.We studies the Dwork pencil of quintic surfaces here.Using the some tricks and intersection form,we describes the pencil in detail.The pencil has five singular surfaces ex- cept the surface consists of the five coordinate planes.And we determine the only possible lines outside the base locus of the pencil,and the smooth surfaces holding such lines.We get the conclusion that only three classes of surfaces contain additional lines,with the number 35,55 and 75.And none of them are isomorphic to the Fermat quintic surface.
引文
[1]K.Arima and I.Shimada,Zariski-van Kampen method and transcendental lattices of certain singular K3 surfaces,arxiv:0806.3311(preprint).
[2]E.Artal-Bartolo,Sur les couples de Zariski,J.Algebraic Geom.3(1994),no.2,223-247.
[3]S.Boissiere and A.Sarti,Counting lines on surfaces,Ann.Scuola Norm.Sup.Pisa Cl.Sci.6(2007),no.5,39-52.
[4]E.Brieskorn and H.Kn(o|¨)rrer,Plane algebraic curves,Birkh(a|¨)user Verlag,Basel Boston Stuttgart,1986.
[5]A.Degtyarev,Alexander polynomial of an algebraic hypersurface,Preprint LOMI R-11-86(1986).
[6]A.Degtyarev,Classification of surfaces of degree four having a non-simple singular point,Math.USSR-Izv.35(1990),no.3,607-627.
[7]A.Degtyarev,Isotopic classification of complex plane projective curves of degree five,Leningrad Math.J.1(1990),no.4,881-904.
[8]A.Degtyarev,Alexander polynomial of a curve of degree six,,J.Knot Theory Ramifications 3(1994),439-454.
[9]A.Degtyarev,On irreducible sextics with non-abelian fundamental group,Fourth Franco-Japanese Symposium on Singularities,2007(to appear).
[10]A.Degtyarev,On deformations of singular plane sextics,J.Algebraaic Geom.17(2008),no.1,101-135.
[11]A.Degtyarev,Zariski k-plets via dessins d'enfants,arXiv:0710.0279,2008.
[12]J.Carmona Ruber E.Artal Bartolo and J.I.Cogolludo Agustin,On sextic curves with big Milnor number,Trends in Singularities(Anatoly Libgober and Mihai Tibar,eds.),Birkh(a|¨)user Verlag,Basel/Switzerland,2002,pp.1-29.
[13]A.M.Uludag,More Zariski pairs and finite fundamental groups of curve complements,Manuscripta Math.106(2001),no.3,271-277.
[14]J.Harris L.Caporaso and B.Mazur,How many rational points can a curve have?,The moduli space of curves(Texel Island,1994),Progr.Math.,no.129,Birkh(a|¨)user Boston,Boston,MA,1995,pp.13-31.
[15]J.Milnor,Singular Points of Complex Hypersurface,Annals Math.Studies,no.61,Princeton Univ.Press,1968.
[16]D.R.Morrison,Mirror symmetry and rational curves on quintic threefolds:A guide for mathematicans,J.AMS.6(1993),no.1,223-247.
[17]D.R.Morrison and M.Saito,Cremona Transformations and Degrees of Period Maps for K3 surfaces with Ordinary Double Points,Algebraic Geometry,Advanced Studies in Pure Mathematics,no.10,Sendai,1987,pp.477-513.
[18]D.Mumford,Algebraic Geometry I Complex Projective Varieties,Springer-Verlag Berlin Heidelberg New York,1976.
[19]M.Namba,Geometry of projective algebraic curves,Marcel Dekker,New York and Basel,1984.
[20]M.Namba and H.Tsuchihashi,On the Fundamental Groups of Galois Covering Spaces of the Projective Plane,Geometriae Dedicata 105(2004),85-105.
[21]V.V.Nikulin,Integral symmetric bilinear forms and some of their applications,Math.USSR-Izv.14(1980),no.1,103-167.
[22]M.Oka and D.T.Pho,Classification of sextics of torus type,Tokyo J.Math.25(2002),no.2,399-433.
[23]M.Oka and D.T.Pho,Fundamental Group of Sextics of Torus type,Trends in Singularities (Anatoly Libgober and Mihai Tiber,ads.),Birkh(a|¨)user Verlag,Basel/Switzerland,2002,pp.151-180.
[24]S.Yu.Orevkov,A new affine M-sextic,Funkts.Anal.Prilozh 32(1988),no.2,91-94.
[25]D.-T.Pho,Classification of singularities on torus curves of type(2,3),Kodai Math.J.24(2001),259-284.
[26]A.Sarti,Pencils of symmetric surfaces in IP~3,J.Algebra 246(2001),no.1,429-452.
[27]M.Sch(u|¨)tt,Elliptic fibrations of some extremal K3 surfaces,Rocky Mountain J.Math.37(2007),no.2,609-652.
[28]M.Sch(u|¨)tt,Fields of definition of singular K3 surfaces,Commun.Number Theory Phys.1(2007),no.2,307-321.
[29]B.Segre,The maximum number of lines lying on a quartic surface,Quart.J.Math.,Oxford Set.14(1943),86-96.
[30]B.Segre,On arithmetical properties of quartic surfaces,Proc.London Math.Soc.49(1947),no.2,353-395.
[31]I.Shimada,Non-homeomorphic conjugate complete varieties,arXiv:math/0701115v2,2007.
[32]I.Shimada,on Arithemetic Zariski Pairs in Degree 6,Adv.Geom.8(2008),no.2,205-225.
[33]I.Shimada,Lattice Zariski k-ples of Plane sextic curves and Z-splitting curves for double plane sextics,arXiv:0903.3308v1,2009.
[34]D.V.Straten,A quintic hypersurface in IP~4 with 130 nodes,Topology 32(1993),no.4,857-864.
[35]H.Tokunaga,Some examples of Zariski pairs arising from certain elliptic K3-surfaces.Ⅱ.Degtyarev's conjecture,Math.Z.230(1999),no.2,389-400.
[36]T.Urabe,Combinations of rational singularities on plane sextic curves with the sum of milnor numbers less than sixteen,Banach center publications 20(1988),429-456.
[37]J.-G.Yang,Table of the lattice and configurations of reduced plane sextic curves with only simple singularities with milnor number from 11 to 19,seminar lectures,unpublished.
[38]J.-G.Yang,Sextic curves with simple singularities,Tohoku Math.J.48(1996),no.2,203-227.
[39]J.-G.Yang and J.-J.Xie,Discriminantal groups and Zariski pairs of sextic curves,arXiv:0903.2058,2009.
[40]H.Yoshihara,On plane rational curves,Proc.Japan Acad.Ser.A Math.Sci.55(1979),no.4,152-155.
[41]O.Zariski,on the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given Branch Curve,Amer.J.Math.51(1929),no.2,305-328.
[2]E.Artal-Bartolo,Sur les couples de Zariski,J.Algebraic Geom.3(1994),no.2,223-247.
[3]S.Boissiere and A.Sarti,Counting lines on surfaces,Ann.Scuola Norm.Sup.Pisa Cl.Sci.6(2007),no.5,39-52.
[4]E.Brieskorn and H.Kn(o|¨)rrer,Plane algebraic curves,Birkh(a|¨)user Verlag,Basel Boston Stuttgart,1986.
[5]A.Degtyarev,Alexander polynomial of an algebraic hypersurface,Preprint LOMI R-11-86(1986).
[6]A.Degtyarev,Classification of surfaces of degree four having a non-simple singular point,Math.USSR-Izv.35(1990),no.3,607-627.
[7]A.Degtyarev,Isotopic classification of complex plane projective curves of degree five,Leningrad Math.J.1(1990),no.4,881-904.
[8]A.Degtyarev,Alexander polynomial of a curve of degree six,,J.Knot Theory Ramifications 3(1994),439-454.
[9]A.Degtyarev,On irreducible sextics with non-abelian fundamental group,Fourth Franco-Japanese Symposium on Singularities,2007(to appear).
[10]A.Degtyarev,On deformations of singular plane sextics,J.Algebraaic Geom.17(2008),no.1,101-135.
[11]A.Degtyarev,Zariski k-plets via dessins d'enfants,arXiv:0710.0279,2008.
[12]J.Carmona Ruber E.Artal Bartolo and J.I.Cogolludo Agustin,On sextic curves with big Milnor number,Trends in Singularities(Anatoly Libgober and Mihai Tibar,eds.),Birkh(a|¨)user Verlag,Basel/Switzerland,2002,pp.1-29.
[13]A.M.Uludag,More Zariski pairs and finite fundamental groups of curve complements,Manuscripta Math.106(2001),no.3,271-277.
[14]J.Harris L.Caporaso and B.Mazur,How many rational points can a curve have?,The moduli space of curves(Texel Island,1994),Progr.Math.,no.129,Birkh(a|¨)user Boston,Boston,MA,1995,pp.13-31.
[15]J.Milnor,Singular Points of Complex Hypersurface,Annals Math.Studies,no.61,Princeton Univ.Press,1968.
[16]D.R.Morrison,Mirror symmetry and rational curves on quintic threefolds:A guide for mathematicans,J.AMS.6(1993),no.1,223-247.
[17]D.R.Morrison and M.Saito,Cremona Transformations and Degrees of Period Maps for K3 surfaces with Ordinary Double Points,Algebraic Geometry,Advanced Studies in Pure Mathematics,no.10,Sendai,1987,pp.477-513.
[18]D.Mumford,Algebraic Geometry I Complex Projective Varieties,Springer-Verlag Berlin Heidelberg New York,1976.
[19]M.Namba,Geometry of projective algebraic curves,Marcel Dekker,New York and Basel,1984.
[20]M.Namba and H.Tsuchihashi,On the Fundamental Groups of Galois Covering Spaces of the Projective Plane,Geometriae Dedicata 105(2004),85-105.
[21]V.V.Nikulin,Integral symmetric bilinear forms and some of their applications,Math.USSR-Izv.14(1980),no.1,103-167.
[22]M.Oka and D.T.Pho,Classification of sextics of torus type,Tokyo J.Math.25(2002),no.2,399-433.
[23]M.Oka and D.T.Pho,Fundamental Group of Sextics of Torus type,Trends in Singularities (Anatoly Libgober and Mihai Tiber,ads.),Birkh(a|¨)user Verlag,Basel/Switzerland,2002,pp.151-180.
[24]S.Yu.Orevkov,A new affine M-sextic,Funkts.Anal.Prilozh 32(1988),no.2,91-94.
[25]D.-T.Pho,Classification of singularities on torus curves of type(2,3),Kodai Math.J.24(2001),259-284.
[26]A.Sarti,Pencils of symmetric surfaces in IP~3,J.Algebra 246(2001),no.1,429-452.
[27]M.Sch(u|¨)tt,Elliptic fibrations of some extremal K3 surfaces,Rocky Mountain J.Math.37(2007),no.2,609-652.
[28]M.Sch(u|¨)tt,Fields of definition of singular K3 surfaces,Commun.Number Theory Phys.1(2007),no.2,307-321.
[29]B.Segre,The maximum number of lines lying on a quartic surface,Quart.J.Math.,Oxford Set.14(1943),86-96.
[30]B.Segre,On arithmetical properties of quartic surfaces,Proc.London Math.Soc.49(1947),no.2,353-395.
[31]I.Shimada,Non-homeomorphic conjugate complete varieties,arXiv:math/0701115v2,2007.
[32]I.Shimada,on Arithemetic Zariski Pairs in Degree 6,Adv.Geom.8(2008),no.2,205-225.
[33]I.Shimada,Lattice Zariski k-ples of Plane sextic curves and Z-splitting curves for double plane sextics,arXiv:0903.3308v1,2009.
[34]D.V.Straten,A quintic hypersurface in IP~4 with 130 nodes,Topology 32(1993),no.4,857-864.
[35]H.Tokunaga,Some examples of Zariski pairs arising from certain elliptic K3-surfaces.Ⅱ.Degtyarev's conjecture,Math.Z.230(1999),no.2,389-400.
[36]T.Urabe,Combinations of rational singularities on plane sextic curves with the sum of milnor numbers less than sixteen,Banach center publications 20(1988),429-456.
[37]J.-G.Yang,Table of the lattice and configurations of reduced plane sextic curves with only simple singularities with milnor number from 11 to 19,seminar lectures,unpublished.
[38]J.-G.Yang,Sextic curves with simple singularities,Tohoku Math.J.48(1996),no.2,203-227.
[39]J.-G.Yang and J.-J.Xie,Discriminantal groups and Zariski pairs of sextic curves,arXiv:0903.2058,2009.
[40]H.Yoshihara,On plane rational curves,Proc.Japan Acad.Ser.A Math.Sci.55(1979),no.4,152-155.
[41]O.Zariski,on the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given Branch Curve,Amer.J.Math.51(1929),no.2,305-328.