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We find explicit free resolutions for the $scr D$-modules ${scr D} f^s$ and ${scr D}[s] f^s/{scr D}[s] f^{s+1}$, where $f$ is a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi-homogeneous free divisor is Koszul free, which is also proved in this paper
We present versal complex analytic families, over a smooth base and of fibre dimension zero, one, or two, where the discriminant constitutes a free divisor. These families include finite flat maps, versal deformations of reduced curve singularities,
For a smooth variety $Y$ over a perfect field of positive characteristic, the sheaf $D_Y$ of crystalline differential operators on $Y$ (also called the sheaf of $PD$-differential operators) is known to be an Azumaya algebra over $T^*_{Y},$ the cotang
We prove a formula for Chow groups of $Quot$-schemes which resolve degeneracy loci of a map between vector bundles, under expected dimension conditions. This result provides a unified way to understand known formulae for various geometric situations
In this short note, we extend the results of [Alexeev-Orlov, 2012] about Picard groups of Burniat surfaces with $K^2=6$ to the cases of $2le K^2le 5$. We also compute the semigroup of effective divisors on Burniat surfaces with $K^2=6$. Finally, we c
This paper studies the derived category of the Quot scheme of rank $d$ locally free quotients of a sheaf $mathscr{G}$ of homological dimension $le 1$ over a scheme $X$. In particular, we propose a conjecture about the structure of its derived categor