# The construction of low - dimensional projective cluster

Author XieJin
Tutor YangJinGen
School Fudan University
Course Basic mathematics
Keywords Zariski pair K3 Surfaces ADE singularity SINGULAR six curves Computational algebraic geometry Five Surfaces Dwork pencil
CLC TP391.41
Type PhD thesis
Year 2009
Downloads 16
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• Abstract
• Contents

Constructive method is one of the effective methods to solve algebraic geometry problem. Constructive method to study a series of low-dimensional projective cluster a singular plane six algebraic curves, and the five surface contains a lot of straight lines. The first part of the study is that the curve of the flat-six Zariski pair, the Urabe grid Yang, the algorithms and Shimada invariant Laixun find the the the large grid distinction Zariskipair and Zariski a triplet, tie on Shimada We conclude, only simple singularities the flat six curves have the same Configuration, but the the discriminant group order Zariski pair a total of 123 pairs, Zariski triplet total of four groups. The second part is the polynomial structural problems processing greatly Six curves. A singular curve Configuration grid approach, we are concerned about how to get it resolved achieve. In this chapter, we introduce some basic knowledge of the structure of the discriminant local reviews the results of the previous curve constructed polynomial equation, and the problem is concentrated Milnor number equal to 19, contains only two focused irreducible six curve constructed. Is different to the previous methods, the main tools we use three types of Cremona transform, and use of a six-curve Pencil with a pencil in a doublet of a cubic curve to approximate the unknown curve, and to prove that the pencil degradation members in Cremona transformation can split certain straight-line members. As a double rational like pencil by relatively simple algebraic equations to express and solve. With the front work, in addition to the two Configuration, we get above 39 Configuration 37. In this chapter, we are starting from the definition given the definition of the combination of Configuration combined algorithm, which can be defined to double rational transform calculus. The third part is a problem in counting algebraic geometry, starting from the Segre about smooth five surfaces on the maximum number of straight lines, consider constructing a symmetric Dwork pencil contains a lot of straight-line five surfaces We take advantage of the symmetry group and the appropriate combination of skills, portrayed pencil solved inside five singular surfaces (removing the the five coordinate plane). Then the use of the intersection of the straight line form proved bp outer straight line, then it will be with at least two containing straight lines intersect at the base, thus uniquely determined explicit expression of the straight line possible, and where the surfaces. Finally, we show that only contains three types of smooth surfaces outside of a straight-line basis points, the number of straight lines containing 35, 55 and 75, and none of them with the known Fermat five surfaces with structure.

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