M.Sc Thesis | |

M.Sc Student | Lanzat Sergei |
---|---|

Subject | Configuration Spaces and Real Enumerative Geometry |

Department | Department of Mathematics |

Supervisor | PROF. Michael Polyak |

Full Thesis text |

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 *R*^{2} 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 *W _{d} *of all real rational
algebraic curves of degree

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^{*} *R*^{2 }of contact
elements of the plane *R*^{2}.