\titleofpresentation{QF-schemes and quadratic forms over local rings}

\presenter{V. M. Levchuk}

The results of \cite{levchuk-milnor,levchuk-LS-msb} receive a
development. The solution of the construction's task of a normal
view for quadratic forms over a local principal ideal ring $R=2R$
with a $QF$-scheme of order 2 is shown. Recall, that group $G=R ^
{*}/R ^ {*2} $ together with its particular map of subsets from $G
$ refers to $QF$-scheme or the quadratic form scheme of ring $R $.
There are $QF $-schemes with abstract group $G $ of exponent 2 are
considered. We give a combinatorial representation for number of
projective congruence quadrics classes of the projective space
over $R$ with a nilpotent maximal ideal. For the projective planes
the quadric's enumerations up to projective equivalence is given;
we consider also the projective planes over rings with a
non-principal maximal ideal. Some problems for $QF$-schemes are
opened.

{This research is supported by Russian fund of fundamental
researches, grant No. 06-01-00824}.

\begin{ourbiblio}{9}

\bibitem{levchuk-milnor}
{\sl Milnor J., Huismoller D.} \ Symmetric bilinear forms. 1985.

\bibitem{levchuk-Kula}
{\sl Kula M. } \ Fields and quadratic form schemes. -- Ann. Math.
Sil. 1985, 1 (13), 7-22.

\bibitem{levchuk-marshall}
{\sl Marshall M. } \ The elementary type conjecture in quadratic
form theory. -- Contemp. Math. 344 (2004), 275-293.

\bibitem{levchuk-bech-hoff}
{\sl Becher K.J., Hoffmann D.W.} \ Symbol lengths in Milnor
$K$-theory. -- Homology, Homotopy and Applications. 6 (2004),
17-31.

\bibitem{levchuk-LS-msb}
{\sl Levchuk V.�., Starikova �.�.} Quadratic forms of projective
spaces over rings. -- Matem. sbornik. 2006, no. 6, 97-110.

\end{ourbiblio}

\presenteraddress{Krasnoyarsk, Siberian  Federal University}

\email{levchuk@lan.krasu.ru}

