2016/05/31 by Patrick Graf, Karl Schwede
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Generalization #Limit (mathematics) #Limit point #Locus (genetics) #Upper and lower bounds #Zero (linguistics) #math.AC #math.AG
paper · pdf · doi:10.1090/proc/13739
published as Proc. AMS 146 (2018), pp. 473-487
openalex created_date 2016/06/24 · openalex publication_date 2017/03/29 · arxiv created 2017/04/11 · arxiv updated 2020/11/10 · openalex updated_date 2026/08/05
We show that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-jumping numbers of a pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma German a right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">a</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X, \mathfrak a)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in positive characteristic have no limit points whenever the symbolic Rees algebra of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus upper K Subscript upper X"> <mml:semantics> <mml:mrow> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>X</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">-KX</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is finitely generated outside an isolated collection of points. We also give a characteristic zero version of this result, as well as a generalization of the Hartshorne–Speiser–Lyubeznik–Gabber stabilization theorem describing the non-<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-pure locus of a variety.