טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentLanzat Sergei
SubjectConfiguration Spaces and Real Enumerative Geometry
DepartmentDepartment of Mathematics
Supervisor Professor Michael Polyak
Full Thesis textFull thesis text - English Version


Abstract

In the present work we study the following question in real enumerative geometry: given a set P of seven points and an immersed curve G  in the real plane R2  all in general position,  how many real rational plane cubics (genus zero algebraic curves of degree three) pass through these points and tangent to this curve. We count each such cubic with a certain sign, and show that their algebraic number is preserved under regular homotopies of a pair (P,G).

In the first part of the paper we recall some basic facts and results from differential and algebraic topology, as well as from algebraic and enumerative geometry. In particular, we discuss in details the space of algebraic curves, topology of singular curves, topology of families of singular algebraic curves, and the intersection theory on smooth manifolds. We explain the basic idea of the work of  J.-P. Welschinger, where he associated a sign to each real rational algebraic curve of degree d and proved that the algebraic number Wd of all real rational algebraic curves of degree d passing through 3d-1 generic points in the real plane R2 counted with that sign is independent on the position of the points.

In the second part of the paper we explain the meaning of general position in our question. We introduce a new sign, with which we count real rational plane cubics passing through P and tangent to G. We give the statement of the main result and discuss the central example, which will be used in the proof.

In the last part we give a proof of the main theorem. We show that the number we get is the intersection number I (L, S) of an appropriate closed smooth one-dimensional submanifold L with a two-dimensional surface S in the space ST* R2 of contact elements of the plane R2.