Exceptional Locus Research Articles

The augmented base locus in mixed characteristic

Let L be a nef and big line bundle on a scheme X. It is well known that if X is a projective over a field then the augmented base locus and the exceptional base locus agree. This result is extended to projective schemes over arbitrary excellent Noetherian bases, assuming the result holds in characteristic zero. In particular the result holds if the base is a mixed characteristic Dedekind domain or if L is semiample in characteristic 0.

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.

The Bertini irreducibility theorem for higher codimensional slices

In [3], Poonen and Slavov recently developed a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing. In this paper, we extend their work by proving an analogous bound for the dimension of the exceptional locus in the setting of linear subspaces of higher codimensions.

Frankenstein’s Self-Portrait: Politicizing Transsexualism in The Danish Girl

This essay aims to contribute to the contemporary transgender debate by using Lili Elbe’s account of her life in Man Into Woman (1933), David Ebershoff’s novel The Danish Girl (2000) and Tom Hooper’s film adaptation of the latter (2015) as case studies. All three narratives explore biologism and medicalization as being closely aligned with the Frankenstein metaphor in terms of the conceptualization of trans bodies. However, this essay contends that Ebershoff’s novel, although tending towards the much-criticized allegorization of such bodies as the exceptional locus of gender trouble, engages in a subtly political enterprise where an androgynous and liberating third space is made available to transgender identities.

Stringy invariants and toric Artin stacks

Abstract We propose a conjectural framework for computing Gorenstein measures and stringy Hodge numbers in terms of motivic integration over arcs of smooth Artin stacks, and we verify this framework in the case of fantastacks, which are certain toric Artin stacks that provide (nonseparated) resolutions of singularities for toric varieties. Specifically, let $\mathcal {X}$ be a smooth Artin stack admitting a good moduli space $\pi : \mathcal {X} \to X$ , and assume that X is a variety with log-terminal singularities, $\pi $ induces an isomorphism over a nonempty open subset of X and the exceptional locus of $\pi $ has codimension at least $2$ . We conjecture a change-of-variables formula relating the motivic measure for $\mathcal {X}$ to the Gorenstein measure for X and functions measuring the degree to which $\pi $ is nonseparated. We also conjecture that if the stabilisers of $\mathcal {X}$ are special groups in the sense of Serre, then almost all arcs of X lift to arcs of $\mathcal {X}$ , and we explain how in this case (assuming a finiteness hypothesis satisfied by fantastacks) our conjectures imply a formula for the stringy Hodge numbers of X in terms of a certain motivic integral over the arcs of $\mathcal {X}$ . We prove these conjectures in the case where $\mathcal {X}$ is a fantastack.

Subvarieties of quotients of bounded symmetric domains

We present a new criterion for the complex hyperbolicity of a non-compact quotient X of a bounded symmetric domain. For each \(p \ge 1\), this criterion gives a precise condition under which the subvarieties \(V \subset X\) with \(\dim V \ge p\) are of general type, and X is p-measure hyperbolic. Then, we give several applications related to ball quotients, or to the Siegel moduli space of principally polarized abelian varieties. In the case of smooth compactifications of ball quotients, we obtain a conditional upper bound on the dimension on the exceptional locus, assuming the Green–Griffiths–Lang conjecture holds true. As another example of application, we give effective lower bounds for levels l so that the moduli spaces of genus g curves with l-level structures are of general type.

Singularity categories via the derived quotient

Given a noncommutative partial resolution A=EndR(R⊕M) of a Gorenstein singularity R, we show that the relative singularity category ΔR(A) of Kalck–Yang is controlled by a certain connective dga A/LAeA, the derived quotient of Braun–Chuang–Lazarev. We think of A/LAeA as a kind of ‘derived exceptional locus’ of the partial resolution A, as we show that it can be thought of as the universal dga fitting into a suitable recollement. This theoretical result has geometric consequences. When R is an isolated hypersurface singularity, it follows that the singularity category Dsg(R) is determined completely by A/LAeA, even when A has infinite global dimension. Thus our derived contraction algebra classifies threefold flops, even those X→Spec(R) where X has only terminal singularities. This gives a solution to the strongest form of the derived Donovan–Wemyss conjecture, which we further show is the best possible classification result in this singular setting.

Moving Seshadri constants, and coverings of varieties of maximal Albanese dimension

Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L \to X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian etale covers of $X$ are arbitrarily large. As an application, given any integer $k\geq 1$, there exists an abelian etale cover $p\colon X' \to X$ such that the adjoint system $\big|K_{X'} + p^*L \big|$ separates $k$-jets away from the augmented base locus of $p^*L$, and the exceptional locus of the pull-back of the Albanese map of $X$ under $p$.

Extracting non-canonical places

Let (X,B) be a log canonical pair and V be a finite set of divisorial valuations with log discrepancy in [0,1). We prove that there exists a projective birational morphism π:Y→X so that the exceptional locus is divisorial, the exceptional divisors are Q-Cartier, and they correspond to elements of V. We study how two such models are related. As applications, we apply the main theorem to the study of deformations of log canonical singularities. Furthermore, we prove the existence of rationalization for log canonical singularities.

The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism

Abstract We introduce a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing over a finite field. Extending a result of Benoist, we prove that for a morphism $\phi \colon X \to{\mathbb{P}}^n$ such that $X$ is geometrically irreducible and the nonempty fibers of $\phi $ all have the same dimension, the locus of hyperplanes $H$ such that $\phi ^{-1} H$ is not geometrically irreducible has dimension at most ${\operatorname{codim}}\ \phi (X)+1$. We give an application to monodromy groups above hyperplane sections.

Finite formal model of toric singularities

We study the formal neighborhoods at rational non-degenerate arcs of the arc scheme associated with a toric variety. The first main result of this article shows that these formal neighborhoods are generically constant on each Nash component of the variety. Furthermore, using our previous work, we attach to every such formal neighborhood, and in fact to every toric valuation, a minimal formal model (in the class of stable isomorphisms) which can be interpreted as a measure of the singularities of the base-variety. As a second main statement, for a large class of toric valuations, we compute the dimension and the embedding dimension of such minimal formal models, and we relate the latter to the Mather discrepancy. The class includes the strongly essential valuations, that is to say those the center of which is a divisor in the exceptional locus of every resolution of singularities of the variety. We also obtain a similar result for monomial curves.

Singularities of restriction varieties in OG(k, n)

ABSTRACTRestriction varieties in the orthogonal Grassmannian are subvarieties of OG(k,n) defined by rank conditions given by a flag that is not necessarily isotropic with respect to the relevant symmetric bilinear form. In particular, Schubert varieties of Type B and D are examples of restriction varieties. In this paper, we introduce a resolution of singularities for restriction varieties in OG(k,n), and give a description of their singular locus by studying components of the exceptional locus of the resolution.

On the topology of a resolution of isolated singularities

Let $Y$ be a complex projective variety of dimension $n$ with isolated singularities, $\pi:X\to Y$ a resolution of singularities, $G:=\pi^{-1}{\rm{Sing}}(Y)$ the exceptional locus. From Decomposition Theorem one knows that the map $H^{k-1}(G)\to H^k(Y,Y\backslash {\rm{Sing}}(Y))$ vanishes for $k>n$. Assuming this vanishing, we give a short proof of Decomposition Theorem for $\pi$. A consequence is a short proof of the Decomposition Theorem for $\pi$ in all cases where one can prove the vanishing directly. This happens when either $Y$ is a normal surface, or when $\pi$ is the blowing-up of $Y$ along ${\rm{Sing}}(Y)$ with smooth and connected fibres, or when $\pi$ admits a natural Gysin morphism. We prove that this last condition is equivalent to say that the map $H^{k-1}(G)\to H^k(Y,Y\backslash {\rm{Sing}}(Y))$ vanishes for any $k$, and that the pull-back $\pi^*_k:H^k(Y)\to H^k(X)$ is injective. This provides a relationship between Decomposition Theorem and Bivariant Theory.

Vector bundles on proper toric 3-folds and certain other schemes

We show that a proper algebraic n n -dimensional scheme Y Y admits non-trivial vector bundles of rank n n , even if Y Y is non-projective, provided that there is a modification containing a projective Cartier divisor that intersects the exceptional locus in only finitely many points. Moreover, there are such vector bundles with arbitrarily large top Chern number. Applying this to toric varieties, we infer that every proper toric threefold admits such vector bundles of rank three. Furthermore, we describe a class of higher-dimensional toric varieties for which the result applies, in terms of convexity properties around rays.

The augmented base locus of real divisors over arbitrary fields

We show that the augmented base locus coincides with the exceptional locus (i.e., null locus) for any nef $$\mathbb R$$ -Cartier divisor on any scheme projective over a field (of any characteristic). Next we prove a semi-ampleness criterion in terms of the exceptional locus generalizing a result of Keel. We also discuss some problems related to augmented base loci of log divisors.

Geometric Reid’s recipe for dimer models

Crepant resolutions of three-dimensional toric Gorenstein singularities are derived equivalent to noncommutative algebras arising from consistent dimer models. By choosing a special stability parameter and hence a distinguished crepant resolution $$Y$$ , this derived equivalence generalises the Fourier-Mukai transform relating the $$G$$ -Hilbert scheme and the skew group algebra $$\mathbb {C}[x,y,z]*G$$ for a finite abelian subgroup of $$\mathrm{SL }(3,\mathbb {C})$$ . We show that this equivalence sends the vertex simples to pure sheaves, except for the zero vertex which is mapped to the dualising complex of the compact exceptional locus. This generalises results of Cautis–Logvinenko (J Reine Angew Math 636:193–236, 2009) and Cautis–Craw–Logvinenko (J Reine Angew Math arXiv:1205.3110 , 2014) to the dimer setting, though our approach is different in each case. We also describe some of these pure sheaves explicitly and compute the support of the remainder, providing a dimer model analogue of results from Logvinenko (J Algebra 324:2064–2087, 2010).

On the noncommutative geometry of square superpotential algebras

A superpotential algebra is square if its quiver admits an embedding into a two-torus such that the image of its underlying graph is a square grid, possibly with diagonal edges in the unit squares; examples are provided by dimer models in physics. Such an embedding reveals much of the algebras representation theory through a device we introduce called an impression. Let A be a square superpotential algebra, Z its center, and m the maximal ideal at the origin of SpecZ. Using an impression, we•give a classification of all simple A-modules up to isomorphism, and give algebraic and homological characterizations of the simple A-modules of maximal k-dimension;•show that Z is a 3-dimensional normal toric domain and Zm is Gorenstein, by determining transcendence bases and Z-regular sequences; and•show that Am is a noncommutative crepant resolution of Zm, and thus a local Calabi–Yau algebra.A particular class of square superpotential algebras, the Yp,q algebras, is considered in detail. We show that the Azumaya and smooth loci of the centers coincide, and propose that each ramified maximal ideal sitting over the singular locus is the exceptional locus of a blowup shrunk to zero size.

Relations of Normal Bundles and Flips of Small Contractions on Projective Manifolds

Let X be a smooth projective variety over the complex number field. Let f : X → Y be a small contraction, and suppose that each irreducible component Eiof the exceptional locus E of f is a smooth subvariety. Assume that dim E ≤ ½ ( dim X + 1), and the normal bundle [Formula: see text]. Then each Ei≅ Pdim Eior Qdim Ei. Moreover, the flip f+: X+→ Y of f exists.

On families of rational curves in the Hilbert square of a surface

For any smooth surface S, the Hilbert scheme S^[n] parameterizing 0-dimensional length-n subschemes of S is a smooth 2n-dimensional variety whose inner geometry is naturally related to that of S. For instance, if E ⊂ S^[n] is the exceptional divisor—that is, the exceptional locus of the Hilbert–Chow morphism μ: S^[n] -> Sym^n(S) — then irreducible (possibly singular) rational curves not contained in E roughly correspond to irreducible (possibly singular) curves on S with a linear series of degree k and dimension 1 on their normalizations, for some k ≤ n. One of the features of this paper is to show how ideas and techniques from one of the two sides of the correspondence make it possible to shed light on problems naturally arising on the other side. If S is moreover a K3 surface then S^[n] is a hyperkahler manifold, and rational curves play a fundamental role in the study of the (birational) geometry of S^[n].

Geminivirus AL2 and L2 Proteins Suppress Transcriptional Gene Silencing and Cause Genome-Wide Reductions in Cytosine Methylation

Geminiviruses replicate single-stranded DNA genomes through double-stranded intermediates that associate with cellular histone proteins. Unlike RNA viruses, they are subject to RNA-directed methylation pathways that target viral chromatin and likely lead to transcriptional gene silencing (TGS). Here we present evidence that the related geminivirus proteins AL2 and L2 are able to suppress this aspect of host defense. AL2 and L2 interact with and inactivate adenosine kinase (ADK), which is required for efficient production of S-adenosyl methionine, an essential methyltransferase cofactor. We demonstrate that the viral proteins can reverse TGS of a green fluorescent protein (GFP) transgene in Nicotiana benthamiana when overexpressed from a Potato virus X vector and that reversal of TGS by geminiviruses requires L2 function. We also show that AL2 and L2 cause ectopic expression of endogenous Arabidopsis thaliana loci silenced by methylation in a manner that correlates with ADK inhibition. However, at one exceptional locus, ADK inhibition was insufficient and TGS reversal required the transcriptional activation domain of AL2. Using restriction-sensitive PCR and bisulfite sequencing, we showed that AL2-mediated TGS suppression is accompanied by reduced cytosine methylation. Finally, using a methylation-sensitive single-nucleotide extension assay, we showed that transgenic expression of AL2 or L2 causes global reduction in cytosine methylation. Our results provide further evidence that viral chromatin methylation is an important host defense and allow us to propose that as a countermeasure, geminivirus proteins reverse TGS by nonspecifically inhibiting cellular transmethylation reactions. To our knowledge, this is the first report that viral proteins can inhibit TGS.