TY - UNPB

T1 - A Semi-orthogonal Sequence in the Derived Category of the Hilbert Scheme of Three Points

AU - Nikolov, Erik

N1 - 38 pages, comments welcome

PY - 2024/4/19

Y1 - 2024/4/19

N2 - For a smooth projective variety $X$ of dimension $d \geq 5$ over an algebraically closed field $k$ of characteristic zero, it is shown in this paper that the bounded derived category of the Hilbert scheme of three points $X^{[3]}$ admits a semi-orthogonal sequence of length $\binom{d-3}{2}$. Each subcategory in this sequence is equivalent to the derived category of $X$ and realized as the image of a Fourier-Mukai transform along a Grassmannian bundle $\mathbb{G}$ over $X$ parametrizing planar subschemes in $X^{[3]}$. The main ingredient in the proof is the computation of the normal bundle of $\mathbb{G}$ in $X^{[3]}$. An analogous result for generalized Kummer varieties is deduced at the end.

AB - For a smooth projective variety $X$ of dimension $d \geq 5$ over an algebraically closed field $k$ of characteristic zero, it is shown in this paper that the bounded derived category of the Hilbert scheme of three points $X^{[3]}$ admits a semi-orthogonal sequence of length $\binom{d-3}{2}$. Each subcategory in this sequence is equivalent to the derived category of $X$ and realized as the image of a Fourier-Mukai transform along a Grassmannian bundle $\mathbb{G}$ over $X$ parametrizing planar subschemes in $X^{[3]}$. The main ingredient in the proof is the computation of the normal bundle of $\mathbb{G}$ in $X^{[3]}$. An analogous result for generalized Kummer varieties is deduced at the end.

KW - math.AG

U2 - 10.48550/arXiv.2404.12851

DO - 10.48550/arXiv.2404.12851

M3 - Preprint

BT - A Semi-orthogonal Sequence in the Derived Category of the Hilbert Scheme of Three Points

ER -