Projective smoothing of varieties with simple normal crossings and their applications

The algebra-topology seminar covers topics in Algebra and Topology

schedule Date & time
11 Jun 2024 3:00pm - 11 Jun 2024 4:00pm
person Speaker


Jayan Mukherjee (Oklahoma State University)
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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


Seminar Room 1.33, Hanna Neumann Building 145
Science Road, Acton ACT 2601