Exceptional Divisor Research Articles

Lefschetz-type theorems for the effective cone on Hyperkähler varieties

In this paper we show some Lefschetz-type theorems for the effective cone of Hyperkähler varieties. In particular we are able to show that the inclusion of any smooth ample divisor induces an isomorphism of effective cones. Moreover we deduce a similar statement for some effective exceptional divisors, which yields the computation of the effective cone of e.g. projectivized cotangent bundles or some projectivized Lazarsfeld–Mukai bundles.

Reflection Vectors and Quantum Cohomology of Blowups

Let $X$ be a smooth projective variety with a semisimple quantum cohomology. It is known that the blowup $\operatorname{Bl}_{\rm pt}(X)$ of $X$ at one point also has semisimple quantum cohomology. In particular, the monodromy group of the quantum cohomology of $\operatorname{Bl}_{\rm pt}(X)$ is a reflectiongroup. We found explicit formulas for certain generators of the monodromy group of the quantum cohomology of $\operatorname{Bl}_{\rm pt}(X)$ depending only on the geometry of the exceptional divisor.

5d SCFTs from isolated complete intersection singularities

In this paper, we explore the zoo of 5d superconformal field theories (SCFTs) constructed from M-theory on Isolated Complete Intersection Singularities (ICIS). We systematically investigate the crepant resolution of such singularities, and obtain a classification of rank ⩽ 10 models with a smooth crepant resolution and smooth exceptional divisors, as well as a number of infinite sequences with the same smoothness properties. For these models, we study their Coulomb branch properties and compute the flavor symmetry algebra from the resolved CY3 and/or the magnetic quiver. We check the validity of the conjectures relating the properties of the 5d SCFT and the 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFT from IIB superstring on the same singularity. When the 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFT has a Lagrangian quiver gauge theory description, one can obtain the magnetic quiver of the 5d theory by gauging flavor symmetry, which encodes the 5d Higgs branch information. Regarding the smoothness of the crepant resolution and integrality of 4d Coulomb branch spectrum, we find examples with a smooth resolved CY3 and smooth exceptional divisors, but fractional 4d Coulomb branch spectrum. Moreover, we compute the discrete (higher)-symmetries of the 5d/4d SCFTs from the link topology for a few examples.

Resolution and alteration with ample exceptional divisor

Resolution and alteration with ample exceptional divisor

Equality of the Wobbly and Shaky Loci

Abstract Let $X$ be a smooth complex projective curve of genus $g\geq 2$, and let $D\subset X$ be a reduced divisor. We prove that a parabolic vector bundle ${\mathcal{E}}$ on $X$ is (strongly) wobbly, that is, ${\mathcal{E}}$ has a non-zero (strongly) parabolic nilpotent Higgs field, if and only if it is (strongly) shaky, that is, it is in the image of the exceptional divisor of a suitable resolution of the rational map from the (strongly) parabolic Higgs moduli to the vector bundle moduli space, both assumed to be smooth. This solves a conjecture by Donagi–Pantev [ 14] in the parabolic and the vector bundle context. To this end, we prove the stability of strongly very stable parabolic bundles, and criteria for very stability of parabolic bundles.

Necessary Codimension One Components of the Projection of the Jacobian Blow-Up

For a complex analytic function, the exceptional divisor of the jacobian blow-up is of great importance. In this short paper, we show what a lemma from the thesis of Lazarsfeld tells one about the structure of the projections of this exceptional divisor into the base space.

Log canonical foliation singularities on surfaces

AbstractWe give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert–Riemenschneider–type vanishing theorem for foliated surfaces with special log canonical foliation singularities.

Deformations of rational curves on primitive symplectic varieties and applications

We study the deformation theory of rational curves on primitive symplectic varieties and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus in the universal locally trivial deformation. As applications, we extend Markman's deformation invariance of prime exceptional divisors along their Hodge locus to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties which are locally trivial deformations of any moduli space of semistable objects on a projective $K3$ or fibers of the Albanese map of those on an abelian surface. We also present an application to the existence of prime exceptional divisors.

On the Behrend function and the blowup of some fat points

The Behrend function of a C-scheme X is a constructible function νX:X(C)→Z introduced by Behrend, intrinsic to the scheme structure of X. It is a (subtle) invariant of singularities of X, playing a prominent role in enumerative geometry. To date, only a handful of general properties of the Behrend function are known. In this paper, we compute it for a large class of fat points (schemes supported at a single point). We first observe that, if X↪AN is a fat point, νX is the sum of the multiplicities of the irreducible components of the exceptional divisor EXAN in the blowup BlXAN. Moreover, we prove that νX can be computed explicitly through the normalisation of BlXAN.The proofs of our explicit formulas for the Behrend function of a fat point in A2 rely heavily on toric geometry techniques. Along the way, we find a formula for the number of irreducible components of EXA2, where X↪A2 is a fat point such that BlXA2 is normal.

Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces

We consider flags E • = {X ⊃ E ⊃ {q}}, where E is an exceptional divisor defining a non-positive at infinity divisorial valuation νE of a Hirzebruch surface , q a point in E and X the surface given by νE , and determine an analogue of the Seshadri constant for pairs (νE , D), D being a big divisor on . The main result is an explicit computation of the vertices of the Newton-Okounkov bodies of pairs (E •, D) as above, showing that they are quadrilaterals or triangles and distinguishing one case from another.

Numerical properties of exceptional divisors of birational morphisms of smooth surfaces

We make a very detailed analysis of effective divisors whose support is contained in the exceptional locus of a birational morphism of smooth projective surfaces. As an application we extend Miyaoka’s inequality on the number of canonical singularities on a projective normal surface with non-negative Kodaira dimension to the non minimal case, obtaining a slightly better result than known extensions by Megyesi and Langer.

Functorial resolution except for toroidal locus. Toroidal compactification

Let X be any variety in characteristic zero. Let V⊂X be an open subset that has toroidal singularities. We show the existence of a canonical desingularization of X except for V. It is a morphism f:Y→X, which does not modify the subset V and transforms X into a toroidal embedding Y, with singularities extending those on V. Moreover, the exceptional divisor has simple normal crossings on Y.The theorem naturally generalizes the Hironaka canonical desingularization. It does not modify the nonsingular locus V and transforms X into a nonsingular variety Y.The proof uses, in particular, the canonical desingularization of logarithmic varieties recently proved by Abramovich-Temkin-Włodarczyk. It also relies on the established here canonical functorial desingularization of locally toric varieties with an unmodified open toroidal subset. As an application, we show the existence of a toroidal equisingular compactification of toroidal varieties. All the results here can be linked to a simple functorial combinatorial desingularization algorithm developed in this paper.

On 2-Representation Infinite Algebras Arising From Dimer Models

AbstractThe Jacobian algebra arising from a consistent dimer model is a bimodule 3-Calabi–Yau algebra, and its center is a 3-dimensional Gorenstein toric singularity. A perfect matching (PM) of a dimer model gives the degree, making the Jacobian algebra $\mathbb{Z}$-graded. It is known that if the degree zero part of such an algebra is finite dimensional, then it is a 2-representation infinite algebra that is a generalization of a representation infinite hereditary algebra. Internal PMs, which correspond to toric exceptional divisors on a crepant resolution of a 3-dimensional Gorenstein toric singularity, characterize the property that the degree zero part of the Jacobian algebra is finite dimensional. Combining this characterization with the theorems due to Amiot–Iyama–Reiten, we show that the stable category of graded maximal Cohen–Macaulay modules admits a tilting object for any 3-dimensional Gorenstein toric isolated singularity. We then show that all internal PMs corresponding to the same toric exceptional divisor are transformed into each other using the mutations of PMs, and this induces derived equivalences of 2-representation infinite algebras.

5D and 6D SCFTs from $\mathbb{C}^3$ orbifolds

We study the orbifold singularities X=\mathbb{C}^3/\GammaX=ℂ3/Γ where \GammaΓ is a finite subgroup of SU(3)SU(3). M-theory on this orbifold singularity gives rise to a 5d SCFT, which is investigated with two methods. The first approach is via 3d McKay correspondence which relates the group theoretic data of \GammaΓ to the physical properties of the 5d SCFT. In particular, the 1-form symmetry of the 5d SCFT is read off from the McKay quiver of \GammaΓ in an elegant way. The second method is to explicitly resolve the singularity XX and study the Coulomb branch information of the 5d SCFT, which is applied to toric, non-toric hypersurface and complete intersection cases. Many new theories are constructed, either with or without an IR quiver gauge theory description. We find that many resolved Calabi-Yau threefolds, \widetilde{X}X̃, contain compact exceptional divisors that are singular by themselves. Moreover, for certain cases of \GammaΓ, the orbifold singularity \mathbb{C}^3/\Gammaℂ3/Γ can be embedded in an elliptic model and gives rise to a 6d (1,0) SCFT in the F-theory construction. Such 6d theory is naturally related to the 5d SCFT defined on the same singularity. We find examples of rank-1 6d SCFTs without a gauge group, which are potentially different from the rank-1 E-string theory.

Pathologies and liftability of Du Val del Pezzo surfaces in positive characteristic

In this paper, we study pathologies of Du Val del Pezzo surfaces defined over an algebraically closed field of positive characteristic by relating them to their non-liftability to the ring of Witt vectors. More precisely, we investigate the condition (NB): all the anti-canonical divisors are singular, (ND): there are no Du Val del Pezzo surfaces over the field of complex numbers with the same Dynkin type, Picard rank, and anti-canonical degree, (NK): there exists an ample \(\mathbb {Z}\)-divisor which violates the Kodaira vanishing theorem for \(\mathbb {Z}\)-divisors, and (NL): the pair (Y, E) does not lift to the ring of Witt vectors, where Y is the minimal resolution and E is its reduced exceptional divisor. As a result, for each of these conditions, we determine all the Du Val del Pezzo surfaces which satisfy the given one.

Regularising Transformations for Complex Differential Equations with Movable Algebraic Singularities

In a 1979 paper, Okamoto introduced the space of initial values for the six Painlevé equations and their associated Hamiltonian systems, showing that these define regular initial value problems at every point of an augmented phase space, a rational surface with certain exceptional divisors removed. We show that the construction of the space of initial values remains meaningful for certain classes of second-order complex differential equations, and more generally, Hamiltonian systems, where all movable singularities of all their solutions are algebraic poles (by some authors denoted the quasi-Painlevé property), which is a generalisation of the Painlevé property. The difference here is that the initial value problems obtained in the extended phase space become regular only after an additional change of dependent and independent variables. Constructing the analogue of space of initial values for these equations in this way also serves as an algorithm to single out, from a given class of equations or system of equations, those equations which are free from movable logarithmic branch points.

Center problem with characteristic directions and inverse integrating factors

We consider the center-focus problem for analytic families of planar vector fields having a monodromic singularity with characteristic directions. We give a method to compute the center conditions of the family provided there is an inverse integrating factor whose expression in polar coordinates has the exceptional divisor of the first polar blow-up as isolated zero-set. Some non-trivial examples are analyzed.

On conjectures of Itoh and of Lipman on the cohomology of normalized blow-ups

Let (R,𝔪,𝕜) be a three-dimensional Cohen–Macaulay analytically unramified local ring and I an 𝔪-primary R-ideal. Write X=Proj(⊕n∈ℕIn¯tn). We prove some consequences of the vanishing of H2(X,𝒪X), whose length equals the constant term e¯3(I) of the normal Hilbert polynomial of I. Firstly, X is Cohen–Macaulay. Secondly, if the extended Rees ring A := ⊕n∈ℤIn¯tn is not Cohen–Macaulay, and either R is equicharacteristic or I¯=m, then e¯2(I)−lengthRI2¯/II¯≥3; this estimate is proved using Boij–Söderberg theory of coherent sheaves on ℙ𝕜2. The two results above are related to a conjecture of S. Itoh (1992). Thirdly, HE2(X,Im𝒪X)=0 for all integers m, where E is the exceptional divisor in X. Finally, if additionally R is regular and X is pseudorational, then the adjoint ideals In ˜, n≥1 satisfy In ˜=IIn−1 ˜ for every n≥3. The last two results are related to conjectures of J. Lipman (1994).

Simple embeddings of rational homology balls and antiflips

Let $V$ be a regular neighborhood of a negative chain of $2$-spheres (i.e. exceptional divisor of a cyclic quotient singularity), and let $B_{p,q}$ be a rational homology ball which is smoothly embedded in $V$. Assume that the embedding is simple, i.e. the corresponding rational blow-up can be obtained by just a sequence of ordinary blow-ups from $V$. Then we show that this simple embedding comes from the semi-stable minimal model program (MMP) for $3$-dimensional complex algebraic varieties under certain mild conditions. That is, one can find all simply embedded $B_{p,q}$'s in $V$ via a finite sequence of antiflips applied to a trivial family over a disk. As applications, simple embeddings are impossible for chains of $2$-spheres with self-intersections equal to $-2$. We also show that there are (infinitely many) pairs of disjoint $B_{p,q}$'s smoothly embedded in regular neighborhoods of (almost all) negative chains of $2$-spheres. Along the way, we describe how MMP gives (infinitely many) pairs of disjoint rational homology balls $B_{p,q}$ embedded in blown-up rational homology balls $B_{n,a} # \bar{\mathbb{CP}^2}$ (via certain divisorial contractions), and in the Milnor fibers of certain cyclic quotient surface singularities. This generalizes results in [Khodorovskiy-2014], [H. Park-J. Park-D. Shin-2016], [Owens-2017] by means of a uniform point of view.

The Kähler-Ricci flow, holomorphic vector fields and Fano bundles

We study the behavior of the Kähler-Ricci flow on compact manifolds developing finite-time singularities, in particular, when the flow contracts exceptional divisors or collapses the Fano fibers of a holomorphic fiber bundle. We present a technique using holomorphic vector fields to prove estimates related to the work of Song-Weinkove and Fu-Zhang.