Primitive multiple schemes

A primitive multiple scheme is a Cohen–Macaulay Y such that the associated reduced $$X=Y_{\mathrm{red}}$$ smooth, irreducible, and can be locally embedded in smooth variety of dimension $$\dim (X)+1$$ . If n multiplicity Y, there canonical filtration $$X=X_1\subset X_2\subset \cdots \subset X_n=Y$$ , $$X_i$$ i. The simplest example trivial to line bundle L on X: it nth infinitesimal neighborhood X, $$L^*$$ by zero section. Let schemes are obtained taking an open cover $$(U_i)$$ X gluing $$U_i\,{\times }\, \mathbf{Z}_n$$ using automorphisms $$U_{ij}\,{\times leave $$U_{ij}$$ invariant. This leads study sheaf nonabelian groups $$X\,{\times invariant, its first cohomology set. $$n\geqslant 2$$ obstruction extension $$X_n$$ $$n+1$$ which lies second group $$H^2(X,E)$$ suitable vector E X. In this paper we these obstructions parametrization schemes. As show if $$X={{\mathbb {P}}}_m$$ with $$m\geqslant 3$$ all trivial. {P}}}_2$$ only two nontrivial schemes, multiplicities 2 4, not quasi-projective. On other hand, projective over curve, infinite sequences $$ X=X_1\subset X_n\subset X_{n+1}\subset