Derived categories of log del Pezzo surfaces with 1/3(1,1) singularities
Abstract
Let X be a log del Pezzo surface with only 1/3(1,1) singularities. We compare the derived categories of its minimal resolution, canonical stack, and singular surface. For every family in the Corti--Heuberger classification, Orlov's blow-up formula and the Ishii--Ueda correspondence give explicit full exceptional collections on the canonical stacks, of length 12−KX2+4k/3 when X has k singular points. In the one-point index-one cascade, the canonical stacks are blow-ups of the weighted projective stack ℙ(1,1,3) at distinct smooth points and admit explicit full strong exceptional collections of line bundles. For quasismooth hypersurfaces X10⊂ℙ(1,2,3,5), we construct a full strong exceptional collection of $13$ objects with an explicitly computed Euler matrix. The exceptional lattice determines the Brauer group: Br(X)=0 for k≤5, while Br(X)≅ℤ/3 for k=6. We construct semiorthogonal decompositions of Db(coh X) for the one- and three-point cascades and of a Brauer-twisted derived category for the six-point cascade.
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BibTeX
@misc{saltachin2026derived,
title = {{Derived categories of log del Pezzo surfaces with $\frac13(1,1)$ singularities}},
author = {Alex Junior Gomez Saltachin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.18238},
eprint = {2606.18238},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.18238},
}
RIS
TY - UNPB
TI - Derived categories of log del Pezzo surfaces with 1/3(1,1) singularities
AU - Alex Junior Gomez Saltachin
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2606.18238
UR - https://arxiv.org/abs/2606.18238
AB - Let X be a log del Pezzo surface with only 1/3(1,1) singularities. We compare the derived categories of its minimal resolution, canonical stack, and singular surface. For every family in the Corti--Heuberger classification, Orlov's blow-up formula and the Ishii--Ueda correspondence give explicit full exceptional collections on the canonical stacks, of length 12−K_X^2+4k/3 when X has k singular points. In the one-point index-one cascade, the canonical stacks are blow-ups of the weighted projective stack ℙ(1,1,3) at distinct smooth points and admit explicit full strong exceptional collections of line bundles. For quasismooth hypersurfaces X_10⊂ℙ(1,2,3,5), we construct a full strong exceptional collection of $13$ objects with an explicitly computed Euler matrix. The exceptional lattice determines the Brauer group: Br(X)=0 for k≤5, while Br(X)≅ℤ/3 for k=6. We construct semiorthogonal decompositions of D^b(coh X) for the one- and three-point cascades and of a Brauer-twisted derived category for the six-point cascade.
ER -