Canonical Divisor Research Articles

Rationality of weighted hypersurfaces of special degree

Let X⊂P(w0,w1,w2,w3) be a quasismooth well-formed weighted projective hypersurface and let L=lcm(w0,w1,w2,w3). We characterize when X is rational under the assumption that L divides deg⁡(X). Furthermore, we give a new family of normal rational weighted projective hypersurfaces with ample canonical divisor, valid in all dimensions, adding to the list of examples discovered by Kollár. Finally, we determine precisely which affine Pham-Brieskorn threefolds are rational, answering a question of Rajendra V. Gurjar.

Realizability of tropical pluri‐canonical divisors

AbstractConsider a pair consisting of an abstract tropical curve and an effective divisor from the linear system associated to times the canonical divisor for . In this article, we give a purely combinatorial criterion to determine if such a pair arises as the tropicalization of a pair consisting of a smooth algebraic curve over a non‐Archimedean field with algebraically closed residue field of characteristic 0 together with an effective pluri‐canonical divisor. To do so, we introduce tropical normalized covers as special instances of cyclic tropical Hurwitz covers and reduce the realizability problem for pluri‐canonical divisors to the realizability problem for normalized covers. Our main result generalizes the work of Möller–Ulirsch–Werner on realizability of tropical canonical divisors and incorporates the recent progress on compactifications of strata of ‐differentials in the work of Bainbridge–Chen–Gendron–Grushevsky–Möller.

Algebraic fibre spaces with strictly nef relative anti‐log canonical divisor

AbstractLet be a projective klt pair, and a fibration to a smooth projective variety with strictly nef relative anti‐log canonical divisor . We prove that is a locally trivial fibration with rationally connected fibres, and the base is a canonically polarized hyperbolic manifold. In particular, when is a single point, we establish that is rationally connected. Moreover, when and is strictly nef, we prove that is ample, which confirms the singular version of a conjecture by Campana and Peternell for threefolds.

Kawaguchi–Silverman conjecture on automorphisms of projective threefolds

Under the framework of dynamics on normal projective varieties by Kawamata, Nakayama and Zhang, and Hu and Li, we may reduce Kawaguchi–Silverman conjecture for automorphisms [Formula: see text] on normal projective threefolds [Formula: see text] with either the canonical divisor [Formula: see text] is trivial or negative Kodaira dimension to the following two cases: (i) [Formula: see text] is a primitively automorphism of a weak Calabi–Yau threefold, (ii) [Formula: see text] is a rationally connected threefold. And we prove Kawaguchi–Silverman conjecture is true for automorphisms of normal projective varieties [Formula: see text] with the irregularity [Formula: see text]. Finally, we discuss Kawaguchi–Silverman conjecture on normal projective varieties with Picard number two.

Minimal compactifications of the affine plane with nef canonical divisors

We shall consider minimal analytic compactifications of the affine plane with singularities. In previous work, Kojima and Takahashi proved that any minimal analytic compactification of the affine plane, which has at worse log canonical singularities, is a numerical del Pezzo surface (i.e., a normal complete algebraic surface with the numerically ample anti-canonical divisor) and has only rational singularities. In this article, we show that any minimal analytic compactification of the affine plane with the nef canonical divisor has an irrational singularity.

Minimizing CM Degree and Specially K-stable Varieties

Abstract We prove that the degree of the CM line bundle for a normal family over a curve with fixed general fibers is strictly minimized if the special fiber is either a smooth projective manifold with a unique cscK metric or “specially K-stable”, which is a new class we introduce in this paper. This phenomenon, as conjectured by Odaka (cf., [ 46]), is a quantitative strengthening of the separatedness conjecture of moduli spaces of polarized K-stable varieties. The above-mentioned special K-stability implies the original K-stability and a lot of cases satisfy it, for example, K-stable log Fano, klt Calabi-Yau (i.e., $K_{X}\equiv 0$), lc varieties with the ample canonical divisor and uniformly adiabatically K-stable klt-trivial fibrations over curves (cf., [ 27]).

Incidence varieties in the projectivized kth Hodge bundle over curves with rational tails

Over the moduli space of pointed smooth algebraic curves, the projectivized [Formula: see text]th Hodge bundle is the space of [Formula: see text]-canonical divisors. The incidence loci are defined by requiring the [Formula: see text]-canonical divisors to have prescribed multiplicities at the marked points. We compute the classes of the closure of the incidence loci in the projectivized [Formula: see text]th Hodge bundle over the moduli space of curves with rational tails. The classes are expressed as a linear combination of tautological classes indexed by decorated stable graphs with coefficients enumerating appropriate weightings. As a consequence, we obtain an explicit expression for some relations in tautological rings of moduli of curves with rational tails.

The Galois module structure of holomorphic poly-differentials and Riemann-Roch spaces

Suppose X is a smooth projective geometrically irreducible curve over a perfect field k of positive characteristic p. Let G be a finite group acting faithfully on X over k such that G has non-trivial, cyclic Sylow p-subgroups. If E is a G-invariant Weil divisor on X with deg(E)>2g(X)−2, we prove that the decomposition of H0(X,OX(E)) into a direct sum of indecomposable kG-modules is uniquely determined by the class of E modulo G-invariant principal divisors, together with the ramification data of the cover X→X/G. The latter is given by the lower ramification groups and the fundamental characters of the closed points of X that are ramified in the cover. As a consequence, we obtain that if m>1 and g(X)≥2, then the kG-module structure of H0(X,ΩX⊗m) is uniquely determined by the class of a canonical divisor on X/G modulo principal divisors, together with the ramification data of X→X/G. This extends to arbitrary m>1 the m=1 case treated by the first author with T. Chinburg and A. Kontogeorgis. We discuss applications to the tangent space of the global deformation functor associated to (X,G) and to congruences between prime level cusp forms in characteristic 0. In particular, we complete the description of the precise kPSL(2,Fℓ)-module structure of all prime level ℓ cusp forms of even weight in characteristic p=3.

The minimal model program for b-log canonical divisors and applications

We discuss the minimal model program for b-log varieties, which is a pair of a variety and a b-divisor, as a natural generalization of the minimal model program for ordinary log varieties. We show that the main theorems of the log MMP work in the setting of the b-log MMP. If we assume that the log MMP terminates, then so does the b-log MMP. Furthermore, the b-log MMP includes both the log MMP and the equivariant MMP as special cases. There are various interesting b-log varieties arising from different objects, including the Brauer pairs, or “non-commutative algebraic varieties which are finite over their centres.” The case of toric Brauer pairs is discussed in further detail.

On endomorphisms of projective varieties with numerically trivial canonical divisors

Let [Formula: see text] be a klt projective variety with numerically trivial canonical divisor. A surjective endomorphism [Formula: see text] is amplified (respectively, quasi-amplified) if [Formula: see text] is ample (respectively, big) for some Cartier divisor [Formula: see text]. We show that after iteration and equivariant birational contractions, a quasi-amplified endomorphism will descend to an amplified endomorphism. As an application, when [Formula: see text] is Hyperkähler, [Formula: see text] is quasi-amplified if and only if it is of positive entropy. In both cases, [Formula: see text] has Zariski dense periodic points. When [Formula: see text] is an abelian variety, we give and compare several cohomological and geometric criteria of amplified endomorphisms and endomorphisms with countable and Zariski dense periodic points (after an uncountable field extension).

Galois quotients of tropical curves and invariant linear systems

For a map $\varphi : \Gamma \rightarrow \Gamma'$ between tropical curves and an isometric action on $\Gamma$ of a finite group $K$, $\varphi$ is a $K$-Galois covering on $\Gamma'$ if $\varphi$ is a harmonic morphism, the degree of $\varphi$ coincides with the order of $K$ and the action of $K$ induces a transitive action on every fibre. We prove that for a tropical curve $\Gamma$ with an isometric action of a finite group $K$, there exists a rational map, from $\Gamma$ to a tropical projective space, which induces a $K$-Galois covering on the image with proper edge-multiplicities. As an application, we also prove that for a hyperelliptic tropical curve without one valent points and of genus at least two, the invariant linear system of the hyperelliptic involution $\iota$ of the canonical linear system, the complete linear system associated with the canonical divisor, induces an $\langle \iota \rangle$-Galois covering on a tree. This is an analogy of the fact that a compact Riemann surface is hyperelliptic if and only if the canonical map, the rational map induced by the canonical linear system, is a double covering on a projective line $\boldsymbol{P}^1$.

Concurrent lines on del Pezzo surfaces of degree one

Let X X be a del Pezzo surface of degree one over an algebraically closed field, and K X K_X its canonical divisor. The morphism φ \varphi induced by | − 2 K X | |-2K_X| realizes X X as a double cover of a cone in P 3 \mathbb {P}^3 , ramified over a smooth sextic curve. The surface X X contains 240 exceptional curves. We prove the following statements. For a point P P on the ramification curve of φ \varphi , at most sixteen exceptional curves contain P P in characteristic 2 2 , and at most ten in all other characteristics. Moreover, for a point Q Q outside the ramification curve, at most twelve exceptional curves contain Q Q in characteristic 3 3 , and at most ten in all other characteristics. We show that these upper bounds are sharp, except possibly in characteristic 5 outside the ramification curve.

Bogomolov-Sommese vanishing and liftability for surface pairs in positive characteristic

We show that the Bogomolov-Sommese vanishing theorem holds for a log canonical projective surface (X,B) in large characteristic unless the Iitaka dimension of KX+⌊B⌋ is not equal to two. As an application, we prove that a log resolution of a pair of a normal projective surface and a reduced divisor in large characteristic lifts to the ring of Witt vectors when the Iitaka dimension of the log canonical divisor is less than or equal to zero. Moreover, we give explicit and optimal bounds on the characteristic unless their Iitaka dimensions are equal to zero.

On the abundance theorem for numerically trivial canonical divisors in positive characteristic

Abstract In this paper, we prove the abundance theorem for numerically trivial canonical divisors on strongly F-regular varieties, assuming that the geometric generic fibers of the Albanese morphisms are strongly F-regular.

Nearby cycles and semipositivity in positive characteristic

We study restriction of logarithmic Higgs bundles to the boundary divisor and we construct the corresponding nearby-cycles functor in positive characteristic. As applications we prove some strong semipositivity theorems for analogs of complex polarized variations of Hodge structures and their generalizations. This implies, e.g., semipositivity for the relative canonical divisor of a semistable reduction in positive characteristic and it gives some new strong results generalizing semipositivity even for complex varieties.

On the positivity of the dimension of the global sections of adjoint bundle for quasi-polarized manifold with numerically trivial canonical bundle

Let $(X, L)$ denote a quasi-polarized manifold of dimension $n \geq 5$ defined over the field of complex numbers such that the canonical line bundle $K_{X}$ of $X$ is numerically equivalent to zero. In this paper, we consider the dimension of the global sections of $K_{X} + mL$ in this case, and we prove that $h^{0}(K_{X} + mL) > 0$ for every positive integer $m$ with $m \geq n - 3$. In particular, a Beltrametti–Sommese conjecture is true for quasi-polarized manifolds with numerically trivial canonical divisors.

On the Cynk-Hulek criterion for crepant resolutions of double covers

A collection A = { D 1 , … , D n } of divisors on a smooth variety X is an arrangement if the intersection of every subset of A is smooth. We show that, if X is defined over a field of characteristic not equal to 2, a double cover of X ramified on an arrangement has a crepant resolution under additional hypotheses. Namely, we assume that all intersection components that change the canonical divisor when blown up are splayed , a property of the tangent spaces of the components first studied by Faber. This strengthens a result of Cynk and Hulek, which requires a stronger hypothesis on the intersection components. Further, we study the singular subscheme of the union of the divisors in A and prove that it has a primary decomposition where the primary components are supported on exactly the subvarieties which are blown up in the course of constructing the crepant resolution of the double cover.

On a generalized canonical bundle formula for generically finite morphisms

We prove a canonical bundle formula for generically finite morphisms in the setting of generalized pairs (with $\mathbb{R}$-coefficients). This complements Filipazzi's canonical bundle formula for morphisms with connected fibres. It is then applied to obtain a subadjunction formula for log canonical centers of generalized pairs. As another application, we show that the image of an anti-nef log canonical generalized pair has the structure of a numerically trivial log canonical generalized pair. This readily implies a result of Chen--Zhang. Along the way we prove that the Shokurov type convex sets for anti-nef log canonical divisors are indeed rational polyhedral sets.

Nef and abundant divisors, semiampleness and canonical bundle formula

In this paper, we use canonical bundle formulas to prove various generalizations of an old theorem of Kawamata on the semiampleness of nef and abundant log canonical divisors. In particular, we show that for klt pairs [Formula: see text] with [Formula: see text] effective, [Formula: see text] nef, nefness and abundance of [Formula: see text] implies semiampleness. This essentially extends Kawamata’s theorem to the setting of generalized abundance.

Bigness of the tangent bundle of del Pezzo surfaces and D-simplicity

We consider the question of simplicity of a ring $R$ under the action of its ring of differential operators $D_R$. We give examples to show that even when $R$ is Gorenstein and has rational singularities $R$ need not be a simple $D_R$-module; for example, this is the case when $R$ is the homogeneous coordinate ring of a smooth cubic surface. Our examples are homogeneous coordinate rings of smooth Fano varieties, and our proof proceeds by showing that the tangent bundle of such a variety need not be big. We also give a partial converse showing that when $R$ is the homogeneous coordinate ring of a smooth projective variety $X$, embedded by some multiple of its canonical divisor, then simplicity of $R$ as a $D_R$-module implies that $X$ is Fano and thus $R$ has rational singularities.