arXiv subjects

Eric Canton

Publications and source records attributed to Eric Canton.

6 recordsLinked to original sources

arXiv2020

The Fedder action and a simplicial complex of local cohomologies

Let $R$ be a regular ring of prime characteristic $p > 0$, and let $\underline{\mathbf{f}}=f_1,\ldots,f_c$ be a permutable regular sequence of codimension $c\geq 1$. We describe a complex of $R\langle F \rangle$-modules, denoted $\Delta\hspace{-2.65mm}\Delta^\bullet_{\underline{\mathbf{f}}}(R)$, whose terms include $\Delta\hspace{-2.65mm}\Delta^0_{\underline{\mathbf{f}}}(R)=R/\underline{\mathbf{f}}$ equipped with its natural Frobenius action, and $\Delta\hspace{-2.65mm}\Delta^c_{\underline{\mathbf{f}}}(R)=H^c_{\underline{\mathbf{f}}}(R)$ equipped with a Frobenius action we refer to as the Fedder action. We show that $H^i(\Delta\hspace{-2.65mm}\Delta^\bullet_{\underline{\mathbf{f}}}(R))=0$ for all $i<c$, and that $H^c(\Delta\hspace{-2.65mm}\Delta^\bullet_{\underline{\mathbf{f}}}(R))$ is a copy of $H^c_{\underline{\mathbf{f}}}(R)$ equipped with the usual Frobenius action. Using the $\Delta\hspace{-2.65mm}\Delta^\bullet_{\underline{\mathbf{f}}}(R)$ complex, we show that if $I\supseteq \underline{\mathbf{f}}$ is an ideal such that $H^i_I(R)=0$ for $\text{ht}(I)<i<\text{ht}(I)+c$ (which is automatic if $R/I$ is Cohen-Macaulay), then the module $H^{\text{ht}(I/\underline{\mathbf{f}})+c}_{I/\underline{\mathbf{f}}}(R/\underline{\mathbf{f}})$ has Zariski closed support.

math.AC↗

arXiv2017

Berkovich log discrepancies in positive characteristic

We introduce and study a log discrepancy function on the space of semivaluations centered on an integral noetherian scheme of positive characteristic. Our definition shares many properties with the analogue in characteristic zero; we prove that if log resolutions exist, then our definition agrees with previous approaches to log discrepancies of semivaluations that these resolutions. We then apply this log discrepancy to a variety of topics in singularity theory over fields of positive characteristic. Strong F-regularity and sharp $F$-purity of Cartier subalgebras are detected using positivity and non-negativity of log discrepancies of semivaluations, just as Kawamata log terminal and log canonical singularities are defined using divisorial log discrepancies, making precise a long-standing heuristic. We prove, in positive characteristic, several theorems of Jonsson and Mustata in characteristic zero regarding log canonical thresholds of graded sequences of ideals. Along the way, we give a valuation-theoretic proof that asymptotic multiplier ideals are coherent on strongly F-regular schemes.

math.AG↗

arXiv2016

A note on injectivity of Frobenius on local cohomology of global complete intersections

Given a graded complete intersection ideal $J = (f_1, \dots, f_c) \subseteq k[x_0, \dots, x_n] = S$, where $k$ is a field of characteristic $p > 0$ such that $[k:k^p] < \infty$, we show that if $S/J$ has an isolated non-F-pure point then the Frobenius action on top local cohomology $H^{n+1-c}_\mathfrak{m}(S/J)$ is injective in sufficiently negative degrees, and we compute the least degree of any kernel element. If $S/J$ has an isolated singularity, we are also able to give an effective bound on $p$ ensuring the Frobenius action on $H^{n+1-c}_\mathfrak{m}(S/J)$ is injective in all negative degrees, extending a result of Bhatt and Singh in the hypersurface case.

math.AC↗

arXiv2015

On the behavior of singularities at the $F$-pure threshold

We provide a family of examples where the $F$-pure threshold and the log canonical threshold of a polynomial are different, but where $p$ does not divide the denominator of the $F$-pure threshold (compare with an example of \mustata-Takagi-Watanabe). We then study the $F$-signature function in the case where either the $F$-pure threshold and log canonical threshold coincide or where $p$ does not divide the denominator of the $F$-pure threshold. We show that the $F$-signature function behaves similarly in those two cases. Finally, we include an appendix which shows that the test ideal can still behave in surprising ways even when the $F$-pure threshold and log canonical threshold coincide.

math.AC↗

arXiv2015

A Note on Injectivity of Frobenius on Local Cohomology of Hypersurfaces

Let $k$ be a field of characteristic $p > 0$ such that $[k:k^p] < \infty$ and let $f \in R = k[x_0, ..., x_n]$ be homogeneous of degree $d$. We obtain a sharp bound on the degrees in which the Frobenius action on $H^n_\mathfrak{m}(R/fR)$ can be injective when $R/fR$ has an isolated non-F-pure point at $\mathfrak{m}$. As a corollary, we show that if $(R/fR)_\mathfrak{m}$ is not F-pure then $R/fR$ has an isolated non-F-pure point at $\mathfrak{m}$ if and only if the Frobenius action is injective in degrees $\le -n(d-1)$.

math.AC↗

arXiv2012

Relating F-Signature and F-Splitting Ratio of Pairs Using Left-Derivatives

We first relate an approximate $n^{th}$-order left derivative of $s(R, f^t)$ at the F-pure threshold $c$ to the F-splitting ratio $r_F(R, f^c)$. Next, we apply the methods developed by Monsky and Teixeira in their investigation of syzygy gaps and $p$-fractals to obtain uniform convergence of the F-signature when $f$ is a product of distinct linear polynomials in two variables. Finally, we explicitly compute the F-signature function for several examples using Macaulay2 code outlined in the last section of this paper.

math.AC↗