Low-dimensional Projective Varieties Constructive

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.