F-adjunction

By Karl Schwede. 2009

104 citations recorded by the source corpus.

In this paper we study singularities defined by the action of Frobenius in characteristic $p>0$. We prove results analogous to inversion of adjunction along a center of log canonicity. For example, we show that if $X$ is a Gorenstein normal variety then to every normal center of sharp $F$-purity $W \subseteq X$ such that $X$ is $F$-pure at the generic point of $W$, there exists a canonically defined $\bQ$-divisor $\Delta_{W}$ on $W$ satisfying $(K_X)|_W \sim_{\bQ} K_{W} + \Delta_{W}$. Furthermore, the singularities of $X$ near $W$ are"the same"as the singularities of $(W, \Delta_{W})$. As an application, we show that there are finitely many subschemes of a quasi-projective variety that are compatibly split by a given Frobenius splitting. We also reinterpret Fedder's criterion in this context, which has some surprising implications.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bf074001561d1e59d75b3edfa355fb7890c6dbcb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

ON THE BEHAVIOR OF TEST IDEALS UNDER FINITE MORPHISMS (2010, methodology): “pondence (2.5.1) stable under iteration via Frobenius. In other words, for all n≥0, the Q-divisor ∆φn associated to (2.5.2) φn:= φ◦Fe ∗ φ◦F 2e ∗ ◦···◦F (n−1)e ∗ φ∈HomO X (Fne ∗ OX,OX) is also ∆φ. See [Sch09] for further details. One may generalize (2.5.1) to include all Q-divisors ∆ ≥0 on Xsuch that”
Test ideals in non-Q-Gorenstein rings (2009, methodology): “The result then follows immediately from Deﬁnition 3.16, Proposition 4.6 and [Sch09, Proposition 4.7].”
A canonical linear system associated to adjoint divisors in characteristic $p > 0$ (2011, methodology): “ In each case, we utilize Proposition 5.3. Therefore, it is suﬃcient to show that the submodule S0 Z,σ(Z,φ∆ Z ) ⊗ OZ(M) is base-point-free. Note that OZ = σ(Z,φ∆ Z ) by the sharp F-purity hypothesis, [Sch09a]. In case (i), we simply applyTheorem 4.3. For case (ii), we notice that (Z,∆Z) is strongly ”
p −1-Linear Maps in Algebra and Geometry (2012, result): “s to the Cartier module F/τ(F) on Ywhose dimension is strictly less than dimX. This yields the following corollary which was obtained in [KM09] and also independently obtained by the second author in [Sch09]. In the case that X= SpecRand Ris local, proofs of this fact were ﬁrst obtained in [Sha07a] ”
F-signature of pairs and the asymptotic behavior of Frobenius splittings (2011, background): “ebra C1/2div(z) ⊆ CkJzK coincides with the Cartier subalgebra generated by ϕ. Since (kJzK,1 2 div(z)) is F-regular, it immediately follows that PR = hx,yi · Ris the splitting prime by Lemma 2.13, cf. [Sch09a]. Furthermore, from example Example 4.19, we see that rF(R) = rF(R,Cϕ) = rF(R,ϕ) = rF(kJzK, ”
Cartier modules: Finiteness results (2009, background): “We are thankful to Karl Schwede for numerous comments on an earlier draft of this manuscript.”
On the three dimensional minimal model program in positive characteristic (2013, unknown): “See [Schwede09, 9.5] for more background.”