DERIVED LENGTHS OF SOLVABLE GROUPS SATISFYING THE ONE-PRIME HYPOTHESIS. III

Abstract

The purpose of this paper is to extend the result in Park (Park, J. (2001). Birational maps of del Pezzo fibrations. J. Reine Angew. Math. 538:213–221) in the case of del Pezzo fibrations of degree 1. To this end we investigate the anticanonical linear systems of del Pezzo surfaces of degree 1. We then classify all possible effective anticanonical divisors on Gorenstein del Pezzo surfaces of degree 1 with canonical singularities.     

Singular del Pezzo surfaces that are equivariant compactifications

We determine which singular del Pezzo surfaces are equivariant compactifications of \mathbbG_\texta² \mathbb{G}_{\text{a}}^2 , to assist with proofs of Manin’s conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of \mathbbG_\texta {\mathbb{G}_{\text{a}}} ⋊ \mathbbG_\textm {\mathbb{G}_{\text{m}}} . Bibliography: 32 titles.     

Orbifold Gromov-Witten theory of weighted blowups

Consider a compact symplectic sub-orbifold groupoid S of a compact symplectic orbifold groupoid (X, ω). Let X_a be the weight-a blowup of X along S, and D_a = PN_a be the exceptional divisor, where N is the normal bundle of S in X. In this paper we show that the absolute orbifold Gromov-Witten theory of X_α can be effectively and uniquely reconstructed from the absolute orbifold Gromov-Witten theories of X, S and D_α, the natural restriction homomorphism H*_CR(X) → H*_CR(S) and the first Chern class of the tautological line bundle over D_α. To achieve this we first prove similar results for the relative orbifold Gromov-Witten theories of (X_α | D_α) and (N_α | D_α). As applications of these results, we prove an orbifold version of a conjecture of Maulik and Pandharipande (Topology, 2006) on the Gromov-Witten theory of blowups along complete intersections, a conjecture on the Gromov-Witten theory of root constructions and a conjecture on the Leray-Hirsch result for the orbifold Gromov-Witten theory of Tseng and You (J Pure Appl Algebra, 2016).

     

Computing α-Invariants of Singular del Pezzo Surfaces

We prove a new local inequality for divisors on surfaces and utilize it to compute α-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$ , $\mathbb{A}_{2}$ , $\mathbb{A}_{3}$ , $\mathbb{A}_{4}$ , $\mathbb{A}_{5}$ , or $\mathbb{A}_{6}$ are Kähler-Einstein.     

On Singular Varieties Having an Extremal Secant Line

ABSTRACT

Let X be a nondegenerate subvariety of degree d and codimension e in the projective space ? ⁿ . If X is smooth, any multisecant line to X cuts X along a 0-dimensional scheme of length at most d ? e + 1. Moreover, smooth varieties X having a (d ? e + 1)-secant line (an extremal secant line) have been completely classified, extending del Pezzo and Bertini classification of varieties of minimal degree. In this article, we almost completely classify possibly singular varieties having an extremal secant line, without any assumptions on the singularities of X. First, we show that, if e ≠ 2, a multisecant line to X meets X along a 0-dimensional scheme of length at most d ? e + 1. Then, we completely classify singular varieties having a (d ? e + 1)-secant line for e ≠ 3. A partial result is provided in case e = 3.     

A characterization of hyperelliptic Jacobians

In this paper we will prove a criterion for hyperelliptic Jacobians. LetD be a translation invariant vector field on an indecompssable principally polarized abelian variety (i.p.p.a.v.) (X, Θ), letDΘ be the divisor of the sectionDΘ∈H ⁰ (Θ,O(Θ)|Θ). We have that (X, Θ) is the Jacobian of an hyperelliptic curve iff (Theorem 1) all the component ofDΘ are non reduced and the singular locus of Θ has dimension less thang-2. We will prove our theorem by showing that under the above geometrical condition it is possible to construct a Kodomcev-Petviashvili (K.P.) equation which is satisfied by the theta function corresponding to the principal polarization onX.     

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface X _k obtained by blowing up ℂℙ² at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W _k :M _k →ℂ with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of X _k , and give an explicit correspondence between the deformation parameters for X _k and the cohomology class [B+iω]∈H ²(M _k ,ℂ).     

Exceptional del Pezzo Hypersurfaces

We compute global log canonical thresholds of a large class of quasismooth well-formed del Pezzo weighted hypersurfaces in ?(a ₀,a ₁,a ₂,a ₃). As a corollary we obtain the existence of orbifold Kähler-Einstein metrics on many of them, and classify exceptional and weakly exceptional quasismooth well-formed del Pezzo weighted hypersurfaces in ?(a ₀,a ₁,a ₂,a ₃).     

Singularities of generic characteristic polynomials and smooth finite splittings of Azumaya algebras over surfaces

Let k be an algebraically closed field. Let P(X ₁₁, . . . , X _nn , T) be the characteristic polynomial of the generic matrix (X _ij ) over k. We determine its singular locus as well as the singular locus of its Galois splitting. If X is a smooth quasi-projective surface over k and A an Azumaya algebra on X of degree n, using a method suggested by M. Artin, we construct finite smooth splittings for A of degree n over X whose Galois closures are smooth.     

Discriminant loci of ample and spanned line bundles

Let (X,L,V) be a triplet where X is an irreducible smooth complex projective variety, L is an ample and spanned line bundle on X and V⊆H⁰(X,L) spans L. The discriminant locus D(X,V)⊂|V| is the algebraic subset of singular elements of |V|. We study the components of D(X,V) in connection with the jumping sets of (X,V), generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of (X,L,V)) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided.     

Log canonical thresholds of del Pezzo surfaces in characteristic p

The global log canonical threshold of each non-singular complex del Pezzo surface was computed by Cheltsov. The proof used Kollár–Shokurov’s connectedness principle and other results relying on vanishing theorems of Kodaira type, not known to be true in finite characteristic. We compute the global log canonical threshold of non-singular del Pezzo surfaces over an algebraically closed field. We give algebraic proofs of results previously known only in characteristic 0. Instead of using of the connectedness principle we introduce a new technique based on a classification of curves of low degree. As an application we conclude that non-singular del Pezzo surfaces in finite characteristic of degree lower or equal than 4 are K-semistable.     

Anneaux noethériens de hilbert

Résumé Soit D un anneau intègre noethérien. On etudie le radical hilbertien de D (ensemble des èlèménts f tels que D_f est un anneau de Hilbert) et on donne des conditions pour que D tout un anneau de Hilbert, en relation avec son radical dimensionnel. Enfin, on caracterise les anneaux de Hilbert par diverses conditions de chaine. On montre en particulier que D est un anneau de Hilbert si et seulement si, pour tout ideal premier 𝔅 de D[X], de trace 𝔭 dans D, la cohauteur de p dans D est majoree par celle de 𝔅 dans D[X].     

Welschinger invariants of toric Del Pezzo surfaces with nonstandard real structures

The Welschinger invariants of real rational algebraic surfaces are natural analogs of the Gromov-Witten invariants, and they estimate from below the number of real rational curves passing through prescribed configurations of points. We establish a tropical formula for the Welschinger invariants of four toric Del Pezzo surfaces equipped with a nonstandard real structure. Such a formula for real toric Del Pezzo surfaces with a standard real structure (i.e., naturally compatible with the toric structure) was established by Mikhalkin and the author. As a consequence we prove that for any real ample divisor D on a surface Σ under consideration, through any generic configuration of c ₁(Σ)D − 1 generic real points, there passes a real rational curve belonging to the linear system |D|. To Vladimir Igorevich Arnold on the occasion of his 70th birthday     

Complex surfaces with equivalent derived categories

We examine the extent to which a smooth minimal complex projective surface X is determined by its derived category of coherent sheaves (DX). To do this we find, for each such surface X, the set of surfaces Y for which there exists a Fourier-Mukai transform D D(X). Received May 10, 1999; in final form July 21, 1999 / Published online February 5, 2001     

The Riemann-Roch theorem on surfaces with log-terminal singularities

Using the singular Riemann-Roch theorem, we propose a method to construct anticanonical sections on singular del Pezzo surfaces. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 4, pp. 35–42, 2004.     

Kawamata–Viehweg vanishing on rational surfaces in positive characteristic

We prove that the Kawamata–Viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to W ₂(k) of certain log pairs on smooth rational surfaces. As a corollary, the Kawamata–Viehweg vanishing theorem holds on log del Pezzo surfaces in positive characteristic.     

Kac–Moody $${\widetilde{E}_k}$$-bundles over elliptic curves and del Pezzo surfaces with singularities of type A

Given an elliptic curve Σ, flat E _k -bundles over Σ are in one-to-one correspondence with smooth del Pezzo surfaces of degree 9 − k containing Σ as an anti-canonical curve. This correspondence was generalized to Lie groups of any type. In this article, we show that there is a similar correspondence between del Pezzo surfaces of degree 0 with an A _d -singularity containing Σ as an anti-canonical curve and Kac–Moody [(E)\tilde]_k{\widetilde{E}_{k}}-bundles over Σ with k = 8 − d. In the degenerate case where surfaces are rational elliptic surfaces, the corresponding [(E)\tilde]_k{\widetilde{E}_k}-bundles over Σ can be reduced to E _k -bundles.     

Geometric Structures on Orbifolds and Holonomy Representations

An orbifold is a topological space modeled on quotient spaces of a finite group actions. We can define the universal cover of an orbifold and the fundamental group as the deck transformation group. Let G be a Lie group acting on a space X. We show that the space of isotopy-equivalence classes of (G, X)-structures on a compact orbifold is locally homeomorphic to the space of representations of the orbifold fundamental group of to G following the work of Thurston, Morgan, and Lok. This implies that the deformation space of (G, X)-structures on is locally homeomorphic to the character variety of representations of the orbifold fundamental group to G when restricted to the region of proper conjugation action by G.     

A Microlocal Riemann–Hilbert Correspondence

We present a microlocal version of the Riemann–Hilbert correspondence for regular holonomic D-modules. We show that a regular holonomic system of microdifferential equations is associated to a perverse sheaf concentrated in degree 0. Moreover, we show that this perverse sheaf can be recovered from the local system it determines on the complementary of its singular locus. We characterize the classes of perverse sheaves and local systems associated to regular holonomic systems of microdifferential equations.