Patchworking of algebraic varieties

Abstract

Consider the diagram

where Y is a germ of a 1-parameter flat¹ family of algebraic varieties with dim Y≥3, such that the fibres Y _t are reduced irreducible for t#0, and the central fibre Y₀ splits into a few reduced components. Further on, X is a germ of a 1-parameter flat family of subvarieties X _t у Yt for t ∈ (ℂ, 0). When considering the diagram (2.1) over the reals, we assume that X and Y are equipped with a complex conjugation which commutes with the projections, and we restrict the parameter range to t ∈ [0, τ), τ>0, taking the respective preimages in X and Y.

The flatness over a smooth one-dimensional base means that the projection is a proper analytic surjection.