Projective smoothing of varieties with simple normal crossings

In this article, we introduce a new approach to show the existence and smoothing of simple normal crossing varieties in a given projective space. Our approach relates the above to the existence of nowhere reduced schemes called ribbons and their smoothings via deformation theory of morphisms. As a consequence, we prove results on the existence and smoothing of snc subvarieties V PN, with two irreducible components, each of which are Fano varieties of dimension n>2, embedded inside P^N for effective values of N, by the complete linear series of a line bundle H. The general fibers of the resulting one parameter families are either smooth Fano, Calabi-Yau or varieties of general type, depending on the positivity of the canonical divisor of their intersections. An interesting consequence of projective smoothing is that it automatically gives a smoothing of the semi-log-canonical (slc) pair (V, ), where = cH, c < 1, is a rational multiple of a general hyperplane section of H. For threefolds, we are able to give explicit descriptions of the smoothable snc subvarieties due to the classification results of Iskovskikh-Mori-Mukai. In particular, we show the existence of unions V = Y₁ D Y₂ PN, where Yᵢ's are smooth anticanonically (resp. bi-anticanonically) embedded Fano threefolds, intersecting along D, where D is either a del-Pezzo surface or a K3 surface (resp. a smooth surface with ample canonical bundle) and their smoothing in PN to smooth Fano or Calabi-Yau threefolds (resp. to threefolds with ample canonical bundle) for various values of N between 10 and 163. In cases when the general fiber is a smooth Fano or Calabi-Yau threefold, one can choose c such that (V, ) is a Calabi-Yau pair while in all cases c can be chosen so that (V, ) is a stable pair.