De Fernex Research Articles

The Nash problem for torus actions of complexity one

Abstract We solve the equivariant generalized Nash problem for any non-rational normal variety with torus action of complexity one. Namely, we give an explicit combinatorial description of the Nash order on the set of equivariant divisorial valuations on any such variety. Using this description, we positively solve the classical Nash problem in this setting, showing that every essential valuation is a Nash valuation. We also describe terminal valuations and use our results to answer negatively a question of de Fernex and Docampo by constructing examples of Nash valuations which are neither minimal nor terminal, thus illustrating a striking new feature of the class of singularities under consideration.

Higher Derivations of Modules and the Hasse–Schmidt Module

In this paper we revisit Ribenboim’s notion of higher derivations of modules and relate it to the recent work of de Fernex and Docampo on the sheaf of differentials of the arc space. In particular, we derive their formula for the Kähler differentials of the Hasse–Schmidt algebra as a consequence of the fact that the Hasse–Schmidt algebra functors commute.

A class of singularity of arbitrary pairs and log canonicalizations

We define a class of singularity on arbitrary pairs of a normal variety and an effective $\mathbb{R}$-divisor on it, which we call pseudo-lc in this paper. This is a generalization of the usual lc singularity of pairs and log canonical singularity of normal varieties introduced by de Fernex and Hacon. By giving examples of pseudo-lc pairs which are not lc or log canonical in the sense of de Fernex--Hacon's paper, we show that pseudo-lc singularity is a strictly extended notion of those singularities. We prove that pseudo-lc pairs admit a small lc modification. We also discuss a criterion of log canonicity.

THE GAMMA CONSTRUCTION AND ASYMPTOTIC INVARIANTS OF LINE BUNDLES OVER ARBITRARY FIELDS

We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is $F$-finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama’s description of the restricted base locus to klt or strongly $F$-regular varieties over arbitrary fields.

The volume of singular Kähler–Einstein Fano varieties

We show that the anti-canonical volume of an$n$-dimensional Kähler–Einstein$\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely$\operatorname{lct}^{n}\cdot \operatorname{mult}$for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler–Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex–Ein–Mustaţă type inequality holds on its affine cone.

Divisorial Models of Normal Varieties

AbstractWe prove that the canonical ring of a canonical variety in the sense of de Fernex and Hacon is finitely generated. We prove that canonical varieties are Kawamata log terminal (klt) if and only if is finitely generated. We introduce a notion of nefness for non-ℚ-Gorenstein varieties and study some of its properties. We then focus on these properties for non-ℚ-Gorenstein toric varieties.

Singularities on demi-normal varieties

The birational classification of varieties inevitably leads to the study of singularities. The types of singularities that occur in this context have been studied by Mori, Kollar, Reid, and others, beginning in the 1980s with the introduction of the Minimal Model Program. Normal singularities that are terminal, canonical, log terminal, and log canonical, and their non-normal counterparts, are typically studied by using a resolution of singularities (or a semi-resolution), and finding numerical conditions that relate the canonical class of the variety to that of its resolution. In order to do this, it has been assumed that a variety X is has a $${\mathbb {Q}}$$ -Cartier canonical class: some multiple $$mK_X$$ of the canonical class is Cartier. In particular, this divisor can be pulled back under a resolution $$f: Y \rightarrow X$$ by pulling back its local sections. Then one has a relation $$K_Y \sim \frac{1}{m}f^*(mK_X) + \sum a_iE_i$$ . It is then the coefficients of the exceptional divisors $$E_i$$ that determine the type of singularities that belong to X. It might be asked whether this $${\mathbb {Q}}$$ -Cartier hypothesis is necessary in studying singularities in birational classification. de Fernex and Hacon (Compos Math 145:393–414, 2009) construct a boundary divisor $$\Delta $$ for arbitrary normal varieties, the resulting divisor $$K_X + \Delta $$ being $${\mathbb {Q}}$$ -Cartier even though $$K_X$$ itself is not. This they call (for reasons that will be made clear) an m-compatible boundary for X, and they proceed to show that the singularities defined in terms of the pair $$(X,\Delta )$$ are none other than the singularities just described, when $$K_X$$ happens to be $${\mathbb {Q}}$$ -Cartier. Thus, a wider context exists within which one can study singularities of the above types. In the present paper, we extend the results of de Fernex and Hacon (Compos Math 145:393–414, 2009) still further, to include demi-normal varieties without a $${\mathbb {Q}}$$ -Cartier canonical class. Our main result is that m-compatible boundaries exist for demi-normal varieties (Theorem 1.1). This theorem provides a link between the theory of singularities for arbitrary demi-normal varieites (whose canonical class may not $$\hbox {be } {\mathbb {Q}}$$ -Cartier), that theory being developed in the present paper, and the established theory of singularities of pairs.

Ovicidal activity of four bioactive plant extracts against the cattle nematode, Cooperia spp. multi-resistant strain

Veterinaria México OA ISSN: 2448-6760Cite this as:von Son de Fernex E, Alonso Díaz MÁ, Mendoza de Gives P, Valles de la Mora B, Zamilpa A, González Cortasar M. Ovicidal activity of extracts from four plant species against the cattle nematode Cooperia punctata. Veterinaria México OA. 2016;3(2). doi: 10.21753/vmoa.3.2.365.Bioactive plants might represent an alternative for Cooperia punctata control in grazing cattle. The objectives of this study were (1) to assess the ovicidal activity of extracts from 4 plant species against C. punctata, (2) to determine the role of the polyphenols in the plants’ anthelmintic (AH) activity, and (3) to evaluate the best plant extraction procedure when searching for ovicidal activity. The egg hatch assay was used with different extraction procedures, aqueous (AQ), acetone:water (AW) and acetonic (AC), to evaluate the ovicidal activity of Leucaena leucocephala, Gliricidia sepium, Guazuma ulmifolia and Cratylia argentea. Eggs of C. punctata were exposed in quadruplicate to 0.6, 1.2, 2.4, 4.8 and 9.6 mg mL-1 of each plant extract. The roles of the polyphenols were assessed using polyethylene glycol (PEG). The 12 plant extracts inhibited egg hatching in a dose-dependent manner. Best-fit LC50 values were 1.03 ± 0.17 and 7.90 ± 1.19 mg mL-1 for G. sepium-AC and L. leucocephala-AQ, respectively. Differences in AH activity were found among the extraction procedures (P < 0.05). At the highest concentration, L. leucocephala-AQ inhibited more than 50% of C. punctata hatching. The G. sepium-AC extract fully inhibited egg hatching. The addition of polyethylene glycol revealed the role of the polyphenols in the bioactivity of most plant extracts; however, for G. sepium-AC, the polyphenols were not the main bioactive compounds. Overall, acetone:water extraction represented the best extraction procedure to obtain both ovicidal activity and higher yield. The inhibition rates suggested that L. leucocephala and G. sepium should be evaluated as a means of reducing larval density in pastures.Table 1. Lethal concentrations required to inhibit 50% of Cooperia punctata egg hatching (LC50), after a 48-h incubation period with bioactive extracts (mg mL-1).

Ovicidal activity of four bioactive plant extracts against the cattle nematode, Cooperia spp. multi-resistant strain

Veterinaria Mexico OA ISSN: 2448-6760 Cite this as: von Son de Fernex E, Alonso Diaz MA, Mendoza de Gives P, Valles de la Mora B, Zamilpa A, Gonzalez Cortasar M. Ovicidal activity of extracts from four plant species against the cattle nematode Cooperia punctata . Veterinaria Mexico OA. 2016;3(2). doi: 10.21753/vmoa.3.2.365 . Bioactive plants might represent an alternative for Cooperia punctata control in grazing cattle. The objectives of this study were (1) to assess the ovicidal activity of extracts from 4 plant species against C. punctata , (2) to determine the role of the polyphenols in the plants’ anthelmintic (AH) activity, and (3) to evaluate the best plant extraction procedure when searching for ovicidal activity. The egg hatch assay was used with different extraction procedures, aqueous (AQ), acetone:water (AW) and acetonic (AC), to evaluate the ovicidal activity of Leucaena leucocephala , Gliricidia sepium , Guazuma ulmifolia and Cratylia argentea . Eggs of C. punctata were exposed in quadruplicate to 0.6, 1.2, 2.4, 4.8 and 9.6 mg mL -1 of each plant extract. The roles of the polyphenols were assessed using polyethylene glycol (PEG). The 12 plant extracts inhibited egg hatching in a dose-dependent manner. Best-fit LC 50 values were 1.03 ± 0.17 and 7.90 ± 1.19 mg mL -1 for G. sepium -AC and L. leucocephala -AQ, respectively. Differences in AH activity were found among the extraction procedures (P < 0.05). At the highest concentration, L. leucocephala -AQ inhibited more than 50% of C. punctata hatching . The G. sepium -AC extract fully inhibited egg hatching. The addition of polyethylene glycol revealed the role of the polyphenols in the bioactivity of most plant extracts; however, for G. sepium -AC, the polyphenols were not the main bioactive compounds. Overall, acetone:water extraction represented the best extraction procedure to obtain both ovicidal activity and higher yield. The inhibition rates suggested that L. leucocephala and G. sepium should be evaluated as a means of reducing larval density in pastures. Table 1. Lethal concentrations required to inhibit 50% of Cooperia punctata egg hatching (LC 50 ), after a 48-h incubation period with bioactive extracts (mg mL -1 ).

Birational rigidity of complete intersections

We prove that every smooth complete intersection \(X=X_{d_{1}, \ldots , d_{s}}\subset \mathbb {P}^{\sum _{i=1}^{s}d_{i}}\) defined by s hypersurfaces of degree \(d_{1}, \ldots , d_{s}\) is birationally superrigid if \(5s +1\le \frac{2(\sum _{i=1}^{s}d_{i}+1)}{\sqrt{\prod _{i=1}^{s}d_{i}}}\). In particular, X is non-rational and \({{\mathrm{Bir}}}(X)={{\mathrm{Aut}}}(X)\). We also prove birational superrigidity of singular complete intersections with similar numerical condition. These extend the results proved by Tommaso de Fernex.

Some Properties and Examples of Log Terminal+ Singularities

In [6], de Fernex and Hacon started the study of singularities on non-ℚ-Gorenstein varieties using pullbacks of Weil divisors. In [4], the author of this article and Urbinati introduce a new class of singularities, called log terminal+, or simply lt+, which they prove is rather well behaved. In this article we will continue the study of lt+ singularities, and we will show that they can be detected by a multiplier ideal, that they satisfy a Bertini type result, inversion of adjunction, and small deformation invariance, and that they are naturally related to rational singularities. Finally, we will provide a list of examples (all of them with lt+ singularities) of the pathologies that can occur in the study of non-ℚ-Gorenstein singularities.

The jumping coefficients of non-[formula omitted]-Gorenstein multiplier ideals

Let a⊂OX be a coherent ideal sheaf on a normal complex variety X, and let c≥0 be a real number. De Fernex and Hacon associated a multiplier ideal sheaf to the pair (X,ac) which coincides with the usual notion whenever the canonical divisor KX is Q-Cartier. We investigate the properties of the jumping numbers associated to these multiplier ideals.We show that the set of jumping numbers of a pair is unbounded, countable and satisfies a certain periodicity property. We then prove that the jumping numbers form a discrete set of real numbers if the locus where KX fails to be Q-Cartier is zero-dimensional. It follows that discreteness holds whenever X is a threefold with rational singularities.Furthermore, we show that the jumping numbers are rational and discrete if one removes from X a closed subset W⊂X of codimension at least three, which does not depend on a. We also obtain that outside of W, the multiplier ideal reduces to the test ideal modulo sufficiently large primes p≫0.

Emotions and adulterous love in 19th century Paris: The letters of Adèle Schunck and Aimé Guyet de Fernex (1824–1849)

In Paris, between 1824 and 1849, Adèle Schunck and Aimé Guyet de Fernex had an illicit relationship in perfect secrecy. This article discusses the place of emotion in the 1500 letters of the two lovers. Analysis, in their words, of the experience and epistolary expression of feelings of love and of the deep emotional turmoil that this generates, leads to describe them as characteristic figures of the Romantic rhetoric. Because of society's disapproval of their love, they are not allowed to express their emotions openly: hence, it is in Nature that the lovers find a link to their feelings.

Comparing multiplier ideals to test ideals on numerically ℚ‐Gorenstein varieties

We show that the reduction to positive characteristic of the multiplier ideal in the sense of de Fernex and Hacon agrees with the test ideal for infinitely many primes, assuming that the variety is numerically ‐Gorenstein. It follows, in particular, that this reduction property holds in dimension 2 for all normal surfaces.

Holomorphic arcs on singularities

The study of formal arcs was initiated by Nash in a 1967 preprint. Arc spaces of smooth varieties have a rather transparent structure but difficult problems arise for arcs passing through singularities. The Nash conjecture on the irreducible components of such arc spaces was proved for surfaces by Fernandez de Bobadilla and Pe Pereira and for toric singularities by Ishii and Kollar, but counter examples were found in higher dimensions by Ishii and Kollar, de Fernex and Johnson and Kollar. Here we start the study of holomorphic arcs; these are holomorphic maps from the closed unit disk to a complex analytic space. As one expects, there is not much conceptual difference between the set of formal arcs and the set of holomorphic arcs since every formal arc can be approximated by holomorphic arcs. However, a formal deformation of an arc is a much more local object than a holomorphic deformation. Thus, in many cases, the space of holomorphic arcs has infinitely many connected components while the space of formal arcs always has only finitely many. For a complex analytic space we define two variants—the space of arcs and the space of short arcs—and we study their connected components. For short arcs we obtain complete answers for surface singularities and for isolated quotient singularities in all dimensions.

The Nash problem and its solution: a survey

The goal of this survey is to give a historical overview of the Nash Problem of arcs in arbitrary dimension, as well as its solution. This problem was stated by J. Nash around 1963 and has been an important subject of research in singularity theory. In dimension two the problem has been solved affirmatively by J. Fernandez de Bobadilla and M. Pe Pereira in 2011. In 2002 S. Ishii and J. Kollar gave a counterexample in dimension four and higher, and in May 2012 T. de Fernex settled (negatively) the last remaining case - that of dimension three. After some history, we give an outline of the solution of the Nash problem for surfaces by Fernandez de Bobadilla and Pe Pereira. We end this survey with the latest series of counterexamples, as well as the Revised Nash problem, both due to J. Johnson and J. Kollar.

Geometric nullstellensatz and symbolic powers on arbitrary varieties

In recent years, a multiplier ideal defined on arbitrary varieties, so called Mather–Jacobian multiplier ideal, has been developed independently by Ein–Ishii–Mustata and de Fernex–Docampo. With this new tool, we have a chance of extending some classical results proved in nonsingular case to arbitrary varieties to establish their general forms. In this paper, we first extend a result of geometric nullstellensatz due to Ein–Lazarsfeld in nonsingular case to any projective varieties. Then we prove a result on comparison of symbolic powers with ordinary powers on any varieties, which extends results of Ein–Lazarsfeld–Smith and Hochster–Huneke.

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.

The volume of an isolated singularity

We introduce a notion of volume of a normal isolated singularity that generalizes Wahl’s characteristic number of surface singularities to arbitrary dimensions. We prove a basic monotonicity property of this volume under finite morphisms. We draw several consequences regarding the existence of noninvertible finite endomorphisms fixing an isolated singularity. Using a cone construction, we deduce that the anticanonical divisor of any smooth projective variety carrying a noninvertible polarized endomorphism is pseudoeffective. Our techniques build on Shokurov’s b-divisors. We define the notions of nef Weil b-divisors and of nef envelopes of b-divisors. We relate the latter to the pullback of Weil divisors introduced by de Fernex and Hacon. Using the subadditivity theorem for multiplier ideals with respect to pairs recently obtained by Takagi, we carry over to the isolated singularity case the intersection theory of nef Weil b-divisors formerly developed by Boucksom, Favre, and Jonsson in the smooth case.

HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES

In this paper we study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces.We introduce a motivic Chern class transformationmCy: K0( var /X) → G0(X) ⊗ ℤ[y], which generalizes the total λ-class λy(T*X) of the cotangent bundle to singular spaces. Here K0( var /X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G0(X) is the Grothendieck group of coherent sheaves of [Formula: see text]-modules. A first construction of mCyis based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mCyis based on some results from the theory of algebraic mixed Hodge modules due to M. Saito.We define a natural transformation Ty*: K0( var /X) → H*(X) ⊗ ℚ[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Ty*is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y = -1), the Todd class transformation in the singular Riemann-Roch theorem of Baum–Fulton–MacPherson (for y = 0) and the L-class transformation of Cappell-Shaneson (for y = 1).We also explain the relation among the "stringy version" of our characteristic classes, the elliptic class of Borisov–Libgober and the stringy Chern classes of Aluffi and De Fernex–Lupercio–Nevins–Uribe.All our results can be extended to varieties over a base field k of characteristic 0.