Log Canonical Singularities Research Articles

Log-canonical degenerations of del Pezzo surfaces in Q-Gorenstein families

We classify del Pezzo surfaces of Picard number one with log canonical singularities admitting Q-Gorenstein smoothings.

Anticanonical system of Fano fivefolds

AbstractWe show that any Fano fivefold with canonical Gorenstein singularities has an effective anticanonical divisor. Moreover, if a general element of the anticanonical system is reduced, then it has canonical singularities.

Characterization of 2-dimensional normal Mather-Jacobian log canonical singularities

In this paper we characterize 2-dimensional normal Mather-Jacobian log canonical singularities which are not complete intersections. We prove that a 2-dimensional normal singularity which is not a complete intersection is a Mather-Jacobian log canonical singularity if and only if it is a toric singularity with embedding dimension 4.

A transcendental approach to injectivity theorem for log canonical pairs

In this paper, we study transcendental aspects of the cohomology groups of adjoint bundles of log canonical pairs, aiming to establish an analytic theory for log canonical singularities. As a result, in the case of purely log terminal pairs, we give an analytic proof of the injectivity theorem originally proved by the Hodge theory. Our method is based on the theory of harmonic integrals and the L2-method for the dbar-equation, and it enables us to generalize the injectivity theorem to the complex analytic setting.

On toric face rings I

We introduce the class of weakly log canonical singularities, a natural generalization of semi-log canonical singularities. Toric varieties (associated to toric face rings, possibly non-normal or reducible) which have weakly (semi-) log canonical singularities are classified. In the toric case, we discuss residues to lc centers of codimension one or higher.

Pull-back of Quasi-Log Structures

We prove that the pull-back of a quasi-log scheme by a smooth quasi-projective morphism has a natural quasi-log structure. We treat an application to log Fano pairs. This paper also contains a proof by Kento Fujita of the simple connectedness of log Fano pairs with only log canonical singularities.

Singularités canoniques et actions horosphériques

Let G be a connected reductive linear algebraic group. We consider the normal G-varieties with horospherical orbits. In this short note, we provide a criterion to determine whether these varieties have at most canonical, log canonical or terminal singularities in the case where they admit an algebraic curve as rational quotient. This result seems to be new in the special setting of torus actions with general orbits of codimension 1. For the given G-variety X, our criterion is expressed in terms of a weight function ωX that is constructed from the set of G-invariant valuations of the function field k(X). In the log terminal case, the generating function of ωX coincides with the stringy motivic volume of X. As an application, we discuss the case of normal k⋆-surfaces.

KSBA surfaces with elliptic quotient singularities, π1 = 1, p g = 0, and K 2 = 1, 2

Among log canonical surface singularities, those which have a rational homology disk smoothing are the cyclic quotient singularities \(\frac{1}{{{n^2}}}\left( {1,na - 1} \right)\) with gcd(a, n) = 1, and three distinguished elliptic quotient singularities. We show the existence of smoothable KSBA normal surfaces with π1 = 1, p g = 0, and K 2 = 1, 2 for each of these three singularities. We also give a list of new (and old) normal surface singularities in smoothable KSBA surfaces for invariants π1 = 1, p g = 0, and K 2 = 1, 2, 3, 4.

On log canonical rational singularities

We prove that the class of log canonical rational singularities is closed under the basic operations of the minimal model program. We also give some supplementary results on the minimal model program for log canonical surfaces.

Castelnuovo–Mumford regularity bounds for singular surfaces

We prove the regularity conjecture, namely Eisenbud–Goto conjecture, for a normal surface with rational, Gorenstein elliptic and log canonical singularities. Along the way, we bound the regularity for a dimension zero scheme by its Loewy length and for a curve allowing embedded or isolated point components by its arithmetic degree.

Singularities of varieties admitting an endomorphism

Let $$X$$ be a normal variety such that $$K_X$$ is $$\mathbb {Q}$$ -Cartier, and let $$f:X \rightarrow X$$ be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that are not log-canonical and the dynamics of the endomorphism $$f$$ . As a consequence we prove that if $$X$$ is projective and $$f$$ polarised, then $$X$$ has at most log-canonical singularities.

An optimal extension theorem for 1-forms and the Lipman–Zariski conjecture

Let X be a normal variety. Assume that for some reduced divisor D \subset X , logarithmic 1-forms defined on the snc locus of (X, D) extend to a log resolution \widetilde X \to X as logarithmic differential forms. We prove that then the Lipman–Zariski conjecture holds for X . This result applies in particular if X has log canonical singularities. Furthermore, we give an example of a 2-form defined on the smooth locus of a three-dimensional log canonical pair (X, \emptyset) which acquires a logarithmic pole along an exceptional divisor of discrepancy zero, thereby improving on a similar example of Greb, Kebekus, Kovács and Peternell.

Nakamaye’s theorem on log canonical pairs

We generalize Nakamaye's description, via intersection theory, of the augmented base locus of a big and nef divisor on a normal pair with log-canonical singularities or, more generally, on a normal variety with non-lc locus of dimension at most 1. We also generalize Ein-Lazarsfeld-Mustata-Nakamaye-Popa's description, in terms of valuations, of the subvarieties of the restricted base locus of a big divisor on a normal pair with klt singularities.

Almost Étale resolution of foliations

In the general, this article addresses the nature of, and relation between, canonical and log-canonical singularities of foliations by curves. The case of equivariance under a finite group action has several peculiarities, amongst which are examples from dimension 3 onward where no birational modification can be both Gorenstein and log-canonical. The article in the specific, i.e. a functorial resolution algorithm for foliated 3-folds in the 2-category of Deligne-Mumford champs, is thus not only optimal but natural.

On the behavior of test ideals under finite morphisms

We derive precise transformation rules for test ideals under an arbitrary finite surjective morphism π : Y → X \pi \colon Y \to X of normal varieties in prime characteristic p > 0 p > 0 . Specifically, given a ℚ -divisor Δ X \Delta _{X} on X X and any O X \mathcal {O}_{X} -linear map T : K ( Y ) → K ( X ) \mathfrak {T} \colon K(Y) \to K(X) , we associate a ℚ -divisor Δ Y \Delta _{Y} on Y Y such that T ( π ∗ τ ( Y ; Δ Y ) ) = τ ( X ; Δ X ) \mathfrak {T} ( \pi _{*}\tau (Y;\Delta _{Y})) = \tau (X;\Delta _{X}) . When π \pi is separable and T = Tr Y / X \mathfrak {T} = \operatorname {Tr}_{Y/X} is the field trace, we have Δ Y = π ∗ Δ X − Ram π \Delta _{Y} = \pi ^{*} \Delta _{X} - \operatorname {Ram}_{\pi } , where Ram π \operatorname {Ram}_{\pi } is the ramification divisor. If, in addition, Tr Y / X ⁡ ( π ∗ O Y ) = O X \operatorname {Tr}_{Y/X}(\pi _{*}\mathcal {O}_{Y}) = \mathcal {O}_{X} , we conclude that π ∗ τ ( Y ; Δ Y ) ∩ K ( X ) = τ ( X ; Δ X ) \pi _{*}\tau (Y;\Delta _{Y}) \cap K(X) = \tau (X;\Delta _{X}) and thereby recover the analogous transformation rule to multiplier ideals in characteristic zero. Our main technique is a careful study of when an O X \mathcal {O}_{X} -linear map F ∗ O X → O X F_{*} \mathcal {O}_{X} \to \mathcal {O}_{X} lifts to an O Y \mathcal {O}_{Y} -linear map F ∗ O Y → O Y F_{*} \mathcal {O}_{Y} \to \mathcal {O}_{Y} , and the results obtained about these liftings are of independent interest as they relate to the theory of Frobenius splittings. In particular, again assuming Tr Y / X ⁡ ( π ∗ O Y ) = O X \operatorname {Tr}_{Y/X}(\pi _{*}\mathcal {O}_{Y}) = \mathcal {O}_{X} , we obtain transformation results for F F -pure singularities under finite maps which mirror those for log canonical singularities in characteristic zero. Finally, we explore new conditions on the singularities of the ramification locus, which imply that, for a finite extension of normal domains R ⊆ S R \subseteq S in characteristic p > 0 p > 0 , the trace map T : Frac ⁡ S → Frac ⁡ R \mathfrak {T} : \operatorname {Frac} S \to \operatorname {Frac} R sends S S onto R R .

On the -purity of isolated log canonical singularities

AbstractA singularity in characteristic zero is said to be of dense $F$-pure type if its modulo $p$ reduction is locally Frobenius split for infinitely many $p$. We prove that if $x\in X$ is an isolated log canonical singularity with $\mu (x\in X)\leq 2$ (where the invariant $\mu $ is as defined in Definition 1.4), then it is of dense $F$-pure type. As a corollary, we prove the equivalence of log canonicity and being of dense $F$-pure type in the case of three-dimensional isolated $ \mathbb{Q} $-Gorenstein normal singularities.

Mori dream spaces of Calabi–Yau type and log canonicity of Cox rings

Abstract We prove that a Mori dream space over a field of characteristic zero is of Calabi–Yau type if and only if its Cox ring has at worst log canonical singularities. By slightly modifying the arguments we also reprove the characterization of varieties of Fano type by log terminality of their Cox rings.

A transversality theorem for some classical varieties

In 2009, de Fernex and Hacon [10] proposed a generalization of the notion of the singularities to normal varieties that are not Q-Gorenstein. Based on their work, we generalize Kleiman’s transversality theorem to subvarieties with log terminal or log canonical singularities. We also show that some classical varieties, such as generic determinantal varieties, Wdr for general smooth curves, and certain Schubert varieties in G(k,n) are log terminal in de Fernex and Hacon’s notion, and canonical with some suitable boundary in the classical sense.

Supplement to “Normal del Pezzo surfaces of rank one with log canonical singularities” by H. Kojima and T. Takahashi [J. Algebra 360 (2012) 53–70

Kojima and Takahashi (2012) [5] proved that every normal del Pezzo surface of rank one with only rational log canonical singular points has at most five singular points. In this paper, we determine such surfaces with five singular points.

Richardson Varieties have Kawamata Log Terminal Singularities

Let $X^v_w$ be a Richardson variety in the full flag variety $X$ associated to a symmetrizable Kac-Moody group $G$. Recall that $X^v_w$ is the intersection of the finite dimensional Schubert variety $X_w$ with the finite codimensional opposite Schubert variety $X^v$. We give an explicit $\bQ$-divisor $\Delta$ on $X^v_w$ and prove that the pair $(X^v_w, \Delta)$ has Kawamata log terminal singularities. In fact, $-K_{X^v_w} - \Delta$ is ample, which additionally proves that $(X^v_w, \Delta)$ is log Fano. We first give a proof of our result in the finite case (i.e., in the case when $G$ is a finite dimensional semisimple group) by a careful analysis of an explicit resolution of singularities of $X^v_w$ (similar to the BSDH resolution of the Schubert varieties). In the general Kac-Moody case, in the absence of an explicit resolution of $X^v_w$ as above, we give a proof that relies on the Frobenius splitting methods. In particular, we use Mathieu's result asserting that the Richardson varieties are Frobenius split, and combine it with a result of N. Hara and K.-I. Watanabe relating Frobenius splittings with log canonical singularities.