Speaker
Description
An important question in birational and enumerative geometry is that of
contractibility of curves in higher dimensional varieties. Motivated by
this, I will focus on rational curves in Calabi--Yau threefolds: the most
recent results in this direction tie contractibility with non-commutative
geometry and deformation theory, through a local model for the formal
neighborhood of the curve.
In particular, Brown and Wemyss compute the non-commutative deformation
theory of curves whose local model has a specific form (introduced by
Katz). I will give a necessary geometric condition for a curve to admit a
formal neighborhood in Katz form. This is also sufficient for curves whose
normal bundle is $(-1,-1)$ and $(-2,0)$. I will then present several
examples, either new or revisiting construction in the literature through
this lens. This is work in progress, joint with M.\ Wemyss.