On the -purity of isolated log canonical singularities

<jats:title>Abstract</jats:title><jats:p>A singularity in characteristic zero is said to be of <jats:italic>dense</jats:italic> <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X1300715X_inline3" /><jats:tex-math>$F$</jats:tex-math></jats:alternatives></jats:inline-formula>-<jats:italic>pure type</jats:italic> if its modulo <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X1300715X_inline4" /><jats:tex-math>$p$</jats:tex-math></jats:alternatives></jats:inline-formula> reduction is locally Frobenius split for infinitely many <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X1300715X_inline5" /><jats:tex-math>$p$</jats:tex-math></jats:alternatives></jats:inline-formula>. We prove that if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X1300715X_inline6" /><jats:tex-math>$x\in X$</jats:tex-math></jats:alternatives></jats:inline-formula> is an isolated log canonical singularity with <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X1300715X_inline7" /><jats:tex-math>$\mu (x\in X)\leq 2$</jats:tex-math></jats:alternatives></jats:inline-formula> (where the invariant <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X1300715X_inline8" /><jats:tex-math>$\mu $</jats:tex-math></jats:alternatives></jats:inline-formula> is as defined in Definition 1.4), then it is of dense <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X1300715X_inline9" /><jats:tex-math>$F$</jats:tex-math></jats:alternatives></jats:inline-formula>-pure type. As a corollary, we prove the equivalence of log canonicity and being of dense <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X1300715X_inline10" /><jats:tex-math>$F$</jats:tex-math></jats:alternatives></jats:inline-formula>-pure type in the case of three-dimensional isolated <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X1300715X_inline11" /><jats:tex-math>$ \mathbb{Q} $</jats:tex-math></jats:alternatives></jats:inline-formula>-Gorenstein normal singularities.</jats:p>