Abstract

In this talk, we discuss an 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. As a consequence, we prove results on the existence and smoothing of snc subvarieties $V \subset \mathbb{P}^N$, with two irreducible components, each of which are Fano varieties of dimension $n>2$, embedded inside $\mathbb{P}^{N}$ 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 gives a smoothing of the semi-log-canonical (slc) pair $(V, \Delta)$, where $\Delta = cH$, $c < 1$, 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 = $Y_1 \bigcup_D Y_2 \subset \mathbb{P}^N$, where $Y_i$'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 $\mathbb{P}^N$ to smooth Fano or Calabi-Yau threefolds (resp. to threefolds with ample canonical bundle) for various values of $N$ between $10$ and $163$. Such projective degenerations can be used to study the problem of extendability of projective varieties. This is joint work with P. Bangere and F.J. Gallego