Stephen Coughlan


Elliptic Gorenstein projection


The archetype elliptic Gorenstein singularity on a $3$-fold is the vertex of the affine cone over a K3 surface. Given a family of $3$-folds of general type, in good cases one may be able to degenerate to a 3-fold with an elliptic Gorenstein singularity, and then project away from this point to a $3$-fold of general type in a smaller ambient projective space. Used in reverse, this resembles Fano's famous strategy for constructing Fano $3$-folds whose anticanonical embedding is in high codimension: construct the image of some projection and then undo the projection. As ever, the more subtle phenomena begin to arise in codimension $4$, when we try to write $X$ in weighted $\mathbb{P}^7$.
This is joint work with Gavin Brown at Loughborough.