Syzygies of Line Bundles on GIT Quotients
Krishna Hanumanthu, Anwesh Ray

TL;DR
This paper investigates the preservation of syzygy properties of line bundles under GIT quotients, establishing conditions under which the property $N_p$ is maintained, with applications to finite groups and specific cases.
Contribution
It proves that the property $N_p$ of a line bundle descends to the GIT quotient under certain conditions, extending understanding of syzygies in geometric invariant theory.
Findings
$N_p$ property is preserved under descent in GIT quotients given certain conditions.
Finite group actions with $N_p$ and $N_0$ properties imply $N_p$ for the descended line bundle.
Applications to specific cases demonstrate the practical relevance of the theoretical results.
Abstract
Let be an algebraically closed field. Consider a reductive group over . Let be a projective variety over with a -action and let be a very ample -linearized line bundle on . Suppose that descends to the GIT quotient of by . If satisfies the property one can ask if its descent also has property. In this article, we show this is the case under certain conditions. We then apply our results to some cases of interest. As a consequence of our results, we show that if is a finite group and satisfies property and its descent satisfies property then it satisfies property as well under suitable conditions.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Commutative Algebra and Its Applications · Advanced Algebra and Geometry
