2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/149925In this paper we consider orthogonal geometry of the free $\protect\ZZ$-module $K_0(\protect\PP_n)$ with respect to the non-symmetric unimodular bilinear form $$χ(E,F)=\sum (-1)^ν\dim\ext^ν(E,F). $$ We calculate the isometry group of this form and describe invariants of its natural action on $K_0(\protect\PP_n)$. Also we consider some general constructions with non-symmetric unimodular forms. In particular, we discuss orthogonal decomposition of such forms and the action of the braid group on a set of semiorthonormal bases. We formulate a list of natural arithmetical conjectures about semiorthogonal bases of the form $χ$.30 pages, LATEX (version 2.01), 12pt, twoside, font families "eufm" and "msbm" are usedAlgebraic Geometrytext