TY - JOUR

T1 - LOWER BOUNDS ON THE F-PURE THRESHOLD AND EXTREMAL SINGULARITIES

AU - Kadyrsizova, Zhibek

AU - Kenkel, Jennifer

AU - Page, Janet

AU - Singh, Jyoti

AU - Smith, Karen E.

AU - Vraciu, Adela

AU - Witt, Emily E.

N1 - Publisher Copyright:
© 2022 by the author(s) under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0) 977.

PY - 2022

Y1 - 2022

N2 - We prove that if f is a reduced homogeneous polynomial of degree d, then its F-pure threshold at the unique homogeneous maximal ideal is at 1 least d−1. We show, furthermore, that itsF-pure threshold equals 1 if and d−1 only if f ∈ m[q] and d = q +1, where q is a power of p. Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such “extremal singularities”, and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are “extremal”, for example, in terms of the configurations of lines they can contain.

AB - We prove that if f is a reduced homogeneous polynomial of degree d, then its F-pure threshold at the unique homogeneous maximal ideal is at 1 least d−1. We show, furthermore, that itsF-pure threshold equals 1 if and d−1 only if f ∈ m[q] and d = q +1, where q is a power of p. Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such “extremal singularities”, and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are “extremal”, for example, in terms of the configurations of lines they can contain.

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U2 - 10.1090/btran/106

DO - 10.1090/btran/106

M3 - Article

AN - SCOPUS:85131950914

SN - 2330-0000

VL - 9

SP - 977

EP - 1005

JO - Transactions of the American Mathematical Society Series B

JF - Transactions of the American Mathematical Society Series B

IS - 31

ER -