Lower bounds on the F-pure threshold and extremal singularities

We prove that if f f is a reduced homogeneous polynomial of degree alttext="d"> d encoding="application/x-tex">d , then its alttext="upper F"> F encoding="application/x-tex">F -pure threshold at the unique maximal ideal least alttext="StartFraction 1 Over d minus EndFraction"> 1 − encoding="application/x-tex">\frac {1}{d-1} . show, furthermore, equals and only alttext="f element-of German m Superscript left-bracket q right-bracket"> ∈<!-- ∈ <mml:msup> m [ q stretchy="false">] encoding="application/x-tex">f\in \mathfrak m^{[q]} alttext="d plus 1"> = + encoding="application/x-tex">d=q+1 where alttext="q"> encoding="application/x-tex">q power alttext="p"> p encoding="application/x-tex">p Up to linear changes coordinates (over fixed algebraically closed field), we classify such “extremal singularities”, show there most one with isolated singularity. Finally, indicate several ways in which projective hypersurfaces defined by forms are “extremal”, for example, terms configurations lines they can contain.