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Let $Xsubset mathbb{P}^4$ be a terminal factorial quartic $3$-fold. If $X$ is non-singular, $X$ is emph{birationally rigid}, i.e. the classical MMP on any terminal $mathbb{Q}$-factorial projective variety $Z$ birational to $X$ always terminates with $X$. This no longer holds when $X$ is singular, but very few examples of non-rigid factorial quartics are known. In this article, we first bound the local analytic type of singularities that may occur on a terminal factorial quartic hypersurface $Xsubset mathbb{P}^4$. A singular point on such a hypersurface is either of type $cA_n$ ($ngeq 1$), or of type $cD_m$ ($mgeq 4$), or of type $cE_6, cE_7$ or $cE_8$. We first show that if $(P in X)$ is of type $cA_n$, $n$ is at most $7$, and if $(P in X)$ is of type $cD_m$, $m$ is at most $8$. We then construct examples of non-rigid factorial quartic hypersurfaces whose singular loci consist (a) of a single point of type $cA_n$ for $2leq nleq 7$ (b) of a single point of type $cD_m$ for $m= 4$ or $5$ and (c) of a single point of type $cE_k$ for $k=6,7$ or $8$.
In this thesis, I determine a bound on the defect of terminal Gorenstein quartic 3-folds. More generally, I study the defect of terminal Gorenstein Fano 3-folds of Picard rank 1 and genus at least 3. I state a geometric motivation of non Q-factoriality in the case of quartics.
In this note we speculate about the structure of maximal product subvarieties in moduli stacks of Calabi-Yau manifolds. We discuss examples for quintic hypersurfaces in the four dimensional projective space.
This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not Q-factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factor
This paper studies the defect of terminal Gorenstein Fano 3 folds. I determine a bound on the defect of terminal Gorenstein Fano 3-folds of Picard rank 1 that do not contain a plane. I give a general bound for quartic 3-folds and indicate how to stud
We compute the Hochschild-Kostant-Rosenberg decomposition of the Hochschild cohomology of Fano 3-folds. This is the first step in understanding the non-trivial Gerstenhaber algebra structure, and yields some initial insights in the classification of