Anticanonical Divisor Research Articles

Rationally connected threefolds with nef and bad anticanonical divisor

Rationally connected threefolds with nef and bad anticanonical divisor

Wall crossing for K‐moduli spaces of plane curves

AbstractWe construct proper good moduli spaces parametrizing K‐polystable ‐Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anticanonical divisor. We then establish a wall‐crossing framework of these K‐moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient is small, the K‐moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K‐moduli spaces are weighted blow‐ups of Kirwan type. We also describe all wall crossings for degree 4,5,6 and relate the final K‐moduli spaces to Hacking's compactification and the moduli of K3 surfaces.

The wrapped Fukaya category for semi-toric Calabi–Yau

We introduce the wrapped Donaldson–Fukaya category of a (generalized) semi-toric SYZ fibration with Lagrangian section satisfying a tameness condition at infinity. Examples include the Gross fibration on the complement of an anti-canonical divisor in a toric Calabi–Yau 3-fold. We compute the wrapped Floer cohomology of a Lagrangian section and find that it is the algebra of functions on the Hori–Vafa mirror. The latter result is the key step in proving homological mirror symmetry for this case. The techniques developed here allow the construction in general of the wrapped Fukaya category on an open Calabi–Yau manifold carrying an SYZ fibration with nice behavior at infinity. We discuss the relation of this to the algebraic vs analytic aspects of mirror symmetry.

Correction to: Positivity of anticanonical divisor in algebraic fibre spaces

Correction to: Positivity of anticanonical divisor in algebraic fibre spaces

Heights and quantitative arithmetic on stacky curves

Abstract In this paper, we investigate the theory of heights in a family of stacky curves following recent work of Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. We count rational points having bounded ESZ-B height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We also show that when the Euler characteristic of stacky curves is non-positive, the ESZ-B height coming from the anti-canonical divisor class fails to have the Northcott property. We prove that a stacky version of a conjecture of Vojta is equivalent to the $abc$ -conjecture.

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.

A Hilbert Irreducibility Theorem for Integral Points on del Pezzo Surfaces

Abstract We prove that the integral points are potentially Zariski dense in the complement of a reduced effective singular anticanonical divisor in a smooth del Pezzo surface, with the exception of ${{\mathbb {P}}}^{2}$ minus three concurrent lines (for which potential density does not hold). This answers positively a question raised by Hassett and Tschinkel and, combined with previous results, completes the proof of the potential density of integral points for complements of anticanonical divisors in smooth del Pezzo surfaces. We then classify the complements that are simply connected and for these we prove that the set of integral points is potentially not thin, as predicted by a conjecture of Corvaja and Zannier.

The moduli space of G-algebras

Let L be a Galois algebra with Galois group G and let x be a normal element of L. The moduli space X of pairs (L,x) is isomorphic to an open subset of the quotient variety P/G, where P is the projective space of the regular representation of G. We provide a formula for the height of any pair (L,x)∈X(Q) in terms of algebraic invariants of L and x with respect to a natural adelic metric on the anticanonical divisor of P/G.

Towards Landau-Ginzburg models for cominuscule spaces via the exceptional cominuscule family

We present projective Landau-Ginzburg models for the exceptional cominuscule homogeneous spaces OP2=E6sc/P6 and E7sc/P7, known respectively as the Cayley plane and Freudenthal variety. These models are defined on the complement Xcan∨ of an anti-canonical divisor of the “Langlands dual homogeneous spaces” X∨=P∨﹨G∨ in terms of generalized Plücker coordinates, analogous to the canonical models defined for Grassmannians, quadrics and Lagrangian Grassmannians in [20,25,23]. We prove that these models for the exceptional family are isomorphic to the Lie-theoretic mirror models defined in [26] using a restriction to an algebraic torus, also known as the Lusztig torus, as proven in [28]. We also give a cluster structure on C[X∨], prove that the Plücker coordinates form a Khovankii basis for a valuation defined using the Lusztig torus, and compute the Newton-Okounkov body associated to this valuation. Although we present our methods for the exceptional types, they generalize immediately to the members of other cominuscule families.

Admissible subcategories of del Pezzo surfaces

We study admissible subcategories of derived categories of coherent sheaves on del Pezzo surfaces and rational elliptic surfaces. Using a relation between admissible subcategories and anticanonical divisors we prove the following results. First, we classify all admissible subcategories of the projective plane by showing that each is generated by a subcollection of a full exceptional collection. Second, we show that the derived categories of del Pezzo surfaces do not contain any phantom subcategories. This provides first examples of varieties of dimension larger than one that have some nontrivial admissible subcategories, but provably do not contain phantoms. We also prove that any admissible subcategory supported set-theoretically on a smooth (−1)-curve in a surface is generated by some twist of the structure sheaf of that curve.

Fano foliations with small algebraic ranks

In this paper we study the algebraic ranks of foliations on Q-factorial normal projective varieties. We start by establishing a Kobayashi-Ochiai's theorem for Fano foliations in terms of algebraic rank. We then investigate the local positivity of the anti-canonical divisors of foliations, obtaining a lower bound for the algebraic rank of a foliation in terms of Seshadri constant. We describe those foliations whose algebraic rank slightly exceeds this bound and classify Fano foliations on smooth projective varieties attaining this bound. Finally we construct several examples to illustrate the general situation, which in particular allow us to answer a question asked by Araujo and Druel on the generalised indices of foliations.

Equations of mirrors to log Calabi–Yau pairs via the heart of canonical wall structures

AbstractGross and Siebert developed a program for constructing in arbitrary dimension a mirror family to a log Calabi–Yau pair (X, D), consisting of a smooth projective variety X with a normal-crossing anti-canonical divisor D in X. In this paper, we provide an algorithm to practically compute explicit equations of the mirror family in the case when X is obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary, and D is the strict transform of the toric boundary. The main ingredient is the heart of the canonical wall structure associated to such pairs (X, D), which is constructed purely combinatorially, following our previous work with Mark Gross. In the case when we blow up a single hypersurface we show that our results agree with previous results computed symplectically by Aroux–Abouzaid–Katzarkov. In the situation when the locus of blow-up is formed by more than a single hypersurface, due to infinitely many walls interacting, writing the equations becomes significantly more challenging. We provide the first examples of explicit equations for mirror families in such situations.

Enumerative geometry of surfaces and topological strings

This survey covers recent developments on the geometry and physics of Looijenga pairs, namely pairs [Formula: see text] with [Formula: see text] a complex algebraic surface and [Formula: see text] a singular anticanonical divisor in it. I will describe a surprising web of correspondences linking together several a priori distant classes of enumerative invariants associated to [Formula: see text], including the log Gromov–Witten invariants of the pair, the Gromov–Witten invariants of an associated higher dimensional Calabi–Yau variety, the open Gromov–Witten invariants of certain special Lagrangians in toric Calabi–Yau threefolds, the Donaldson–Thomas theory of a class of symmetric quivers, and certain open and closed BPS-type invariants. I will also discuss how these correspondences can be effectively used to provide a complete closed-form solution to the calculation of all these invariants.

On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces

In this paper, we study the relative anti-canonical divisor − K X / Y -K_{X/Y} of an algebraic fiber space ϕ : X → Y \phi \colon X\to Y , and we reveal relations among positivity conditions of − K X / Y -K_{X/Y} , certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana–Cao–Matsumura’s equality on Hacon–McKernan’s question, whose original proof depends on analytics methods. The third result proves that algebraic fiber spaces with semi-ample relative anti-canonical divisor actually have a product structure via the base change by an appropriate finite étale cover of Y Y . Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.

K-stability of Fano threefolds of rank 2 and degree 14 as double covers

We prove that every smooth Fano threefold from the family N\(^{\underline{\textrm{o}}}\)2.8 is K-stable. Such a Fano threefold is a double cover of the blow-up of \(\mathbb {P}^3\) at one point branched along an anti-canonical divisor.

On the fundamental group of open Richardson varieties

We compute the fundamental group of an open Richardson variety in the manifold of complete flags that corresponds to a partial flag manifold. Rietsch showed that these log Calabi-Yau varieties underlie a Landau-Ginzburg mirror for the Langlands dual partial flag manifold, and our computation verifies a prediction of Hori for this mirror. It is log Calabi-Yau as it isomorphic to the complement of the Knutson-Lam-Speyer anti-canonical divisor for the partial flag manifold. We also determine explicit defining equations for this divisor.

Curve classes on Calabi–Yau complete intersections in toric varieties

AbstractWe prove the integral Hodge conjecture for curve classes on smooth varieties of dimension at least three constructed as a complete intersection of ample hypersurfaces in a smooth projective toric variety, such that the anticanonical divisor is the restriction of a nef divisor. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric minimal model program, we prove that in each case, is generated by classes of rational curves.

Fano varieties with large Seshadri constants in positive characteristic

We prove that a Fano variety (with arbitrary singularities) of dimension n in positive characteristic is isomorphic to ℙ n if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than n and classify Fano varieties whose anti-canonical divisors have Seshadri constants n. In characteristic p>5 and dimension 3, we also show that Fano varieties X with Seshadri constants ϵ(-K X ,x)>2+ϵ at some smooth point x∈X (for some fixed ϵ>0) have bounded anti-canonical degrees.

Measure Preserving Holomorphic Vector Fields, Invariant Anti-Canonical Divisors and Gibbs Stability

Let X be a compact complex manifold whose anti-canonical line bundle — KX is big. We show that X admits no non-trivial holomorphic vector fields if it is Gibbs stable (at any level). The proof is based on a vanishing result for measure preserving holomorphic vector fields on X of independent interest. As an application it shown that, in general, if — KX is big, there are no holomorphic vector fields on X that are tangent to a non-singular irreducible anti-canonical divisor S on X. More generally, the result holds for varieties with log terminal singularities and log pairs. Relations to a result of Berndtsson about generalized Hamiltonians and coercivity of the quantized Ding functional are also pointed out.

A Plücker coordinate mirror for type A flag varieties

We introduce a superpotential for partial flag varieties of type A $A$ . This is a map W : Y ∘ → C $W: Y^\circ \rightarrow \mathbb {C}$ , where Y ∘ $Y^\circ$ is the complement of an anticanonical divisor on a product of Grassmannians. The map W $W$ is expressed in terms of Plücker coordinates of the Grassmannian factors. This construction generalizes the Marsh–Rietsch Plücker coordinate mirror for Grassmannians. We show that in a distinguished cluster chart for Y $Y$ , our superpotential agrees with earlier mirrors constructed by Eguchi–Hori–Xiong and Batyrev–Ciocan-Fontanine–Kim–van Straten. Our main tool is quantum Schubert calculus on the flag variety.