Finite generation of relative log canonical algebras of semi-dlt pairs
Abstract
We prove that the relative log canonical algebra of a semi-dlt pair (X,ΔX) projective over a scheme T is finitely generated, equivalently, that (X,ΔX) admits a relative stable model, provided that the normalization admits a log canonical model over T, that (X,ΔX) admits a stable model over an open dense subscheme T0⊂ T whose exceptional locus does not contain any stratum of the conductor, and that the log centers contained in the conductor have images meeting T0. Our main contribution is a direct proof that avoids Kollár's gluing theory entirely. As a consequence, we recover demi-normal versions of results of Hacon--Xu and Birkar. Furthermore, we derive the existence of certain MMP steps for slc pairs, recovering results of Ambro and Kollár. Finally, these results lay the groundwork for a streamlined proof of the properness of the Kollár--Shepherd-Barron--Alexeev moduli space of stable pairs, to be completed in forthcoming work.
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BibTeX
@misc{landi2026finite,
title = {{Finite generation of relative log canonical algebras of semi-dlt pairs}},
author = {Alberto Landi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36018},
eprint = {2609.36018},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36018},
}
RIS
TY - UNPB
TI - Finite generation of relative log canonical algebras of semi-dlt pairs
AU - Alberto Landi
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.36018
UR - https://arxiv.org/abs/2609.36018
AB - We prove that the relative log canonical algebra of a semi-dlt pair (X,Δ_X) projective over a scheme T is finitely generated, equivalently, that (X,Δ_X) admits a relative stable model, provided that the normalization admits a log canonical model over T, that (X,Δ_X) admits a stable model over an open dense subscheme T^0⊂ T whose exceptional locus does not contain any stratum of the conductor, and that the log centers contained in the conductor have images meeting T^0. Our main contribution is a direct proof that avoids Kollár's gluing theory entirely. As a consequence, we recover demi-normal versions of results of Hacon--Xu and Birkar. Furthermore, we derive the existence of certain MMP steps for slc pairs, recovering results of Ambro and Kollár. Finally, these results lay the groundwork for a streamlined proof of the properness of the Kollár--Shepherd-Barron--Alexeev moduli space of stable pairs, to be completed in forthcoming work.
ER -