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The aim of this paper is to propose a strategy to implement the Minimal Model Program in modern computer algebra systems. [ABSTRACT FROM AUTHOR]

We introduce a novel wall-crossing phenomenon in the space of Bridgeland stability conditions: a wall in the stability space of canonical genus 4 curves that induces non-Q-factorial singularities and hence, it cannot be detected as an operation in the Minimal Model Program of the corresponding moduli space, unlike the case for many surfaces. More precisely, we give an example of a wall-crossing in D b (P 3) such that the wall induces a small contraction of the moduli space of stable objects associated to one of the adjacent chambers, but a divisorial contraction to the other. This significantly complicates the overall picture in this correspondence to applications of stability conditions to algebraic geometry. The full wall-crossing for canonical genus four curves and the geometry are considered in the published paper (Rezaee in Proc LMS 128(1):e12577, 2024); this article is devoted to describe a particularly interesting wall among the walls in Rezaee (Proc LMS 128(1):e12577, 2024) in full details to explain the novel phenomenon. [ABSTRACT FROM AUTHOR]

In this short note we correct a statement contained in the final section of “Birational geometry of irreducible holomorphic symplectic tenfolds of O’Grady type” about the Beauville–Bogomolov square of the twisted theta divisor in the twisted intermediate jacobian fibration of a smooth cubic fourfold. [ABSTRACT FROM AUTHOR]

Let (X , Δ) be a projective, Q -factorial log canonical pair and let L be a pseudoeffective Q -divisor on X such that K X + Δ + L is pseudoeffective. Is there an effective Q -divisor M on X such that K X + Δ + L is numerically equivalent to M? We are not aware of any counterexamples, but the answer is not completely clear even in the case of surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the classification of complex prime Q -Fano 3-folds of anti-canonical codimension 4 which are produced, as weighted complete intersections of appropriate weighted projectivizations of certain affine varieties related with P 1 × P 1 × P 1 -fibrations. Such affine varieties or their appropriate weighted projectivizations are called key varieties for prime Q -Fano 3-folds. We realize that the equations of the key varieties can be described conceptually by Freudenthal triple systems (FTS, for short). The paper consists of two parts. In Part 1, we revisit the general theory of FTS; the main purpose of Part 1 is to derive the conditions of so called strictly regular elements in FTS so as to fit with our description of key varieties. Then, in Part 2, we define several key varieties for prime Q -Fano 3-folds from the conditions of strictly regular elements in FTS. Among other things obtained in Part 2, we show that there exists a 14-dimensional factorial affine variety U A 14 of codimension 4 in an affine 18-space with only Gorenstein terminal singularities, and we construct examples of prime Q -Fano 3-folds of No.20544 in as reported by Altınok et al. (The graded ring database, ) as weighted complete intersections of the weighted projectivization of U A 14 in the weighted projective space P (1 15 , 2 2 , 3) . We also clarify in Part 2 a relation between U A 14 and the G 2 (4) -cluster variety, which is a key variety for prime Q -Fano 3-folds constructed in Coughlan and Ducat (Compos. Math. 156:1873-1914, 2020). [ABSTRACT FROM AUTHOR]

Given a projective or compact Kähler manifold X and a (smooth) hypersurface Y, we study conditions under which X \ Y could be Stein. We apply this in particular to the case when X is the projectivization of the so-called canonical extension of the tangent bundle T M of a projective manifold M with Y being the projectivization of T M itself. [ABSTRACT FROM AUTHOR]

We prove that the quotient of a klt type singularity by a reductive group is of klt type in characteristic 0. In particular, given a klt variety X endowed with the action of a reductive group G and admitting a quasi-projective good quotient X → X / / G , we can find a boundary B on X / / G so that the pair (X / / G , B) is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian Kähler G -manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing n -dimensional K-polystable smooth Fano varieties of volume v has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type. [ABSTRACT FROM AUTHOR]

This paper is concerned about a class of finite time collapsing along continuity method over a compact Kähler manifold. It is shown that along the continuity method the diameter is uniformly bounded and its Gromov–Hausdorff limit is homeomorphic and locally isometric to the regular part of base space. Furthermore, we also prove that the Gromov–Hausdorff limit is unique under the algebraic assumption of the Kähler class by applying the recent work of Song–Tian–Zhang [2019, arXiv:1904.08345]. [ABSTRACT FROM AUTHOR]

Inspired by Schmidt's work on twisted cubics, we study wall crossings in Bridgeland stability, starting with the Hilbert scheme Hilb 2 m + 2 (P 3) parametrizing pairs of skew lines and plane conics union a point. We find two walls. Each wall crossing corresponds to a contraction of a divisor in the moduli space and the contracted space remains smooth. Building on work by Chen–Coskun–Nollet, we moreover prove that the contractions are K-negative extremal in the sense of Mori theory and so the moduli spaces are projective. [ABSTRACT FROM AUTHOR]

In characteristic 0, the Campana-Peternell conjecture claims that the only smooth Fano variety with nef tangent bundle should be homogeneous. In this paper, we study the positive characteristic version of the Campana-Peternell conjecture. In particular, we give an affirmative answer for Fano 4-folds with nef tangent bundle and Picard number greater than one. [ABSTRACT FROM AUTHOR]

Inspired by Schmidt's work on twisted cubics, we study wall crossings in Bridgeland stability, starting with the Hilbert scheme Hilb 2 m + 2 (P 3) parametrizing pairs of skew lines and plane conics union a point. We find two walls. Each wall crossing corresponds to a contraction of a divisor in the moduli space and the contracted space remains smooth. Building on work by Chen–Coskun–Nollet, we moreover prove that the contractions are K-negative extremal in the sense of Mori theory and so the moduli spaces are projective. [ABSTRACT FROM AUTHOR]

We investigate the rationality problem for Q -Fano threefolds of Fano index ≥ 2 . [ABSTRACT FROM AUTHOR]

We give constructions of completions of the affine 3-space into total spaces of del Pezzo fibrations of every degree other than 7 over the projective line. We show in particular that every del Pezzo surface other than P 2 blown-up in one or two points can appear as a closed fiber of a del Pezzo fibration π : X → P 1 whose total space X is a Q -factorial threefold with terminal singularities which contains A 3 as the complement of the union of a closed fiber of π and a prime divisor B h horizontal for π . For such completions, we also give a complete description of integral curves that can appear as general fibers of the induced morphism π ¯ : B h → P 1 . [ABSTRACT FROM AUTHOR]

Let M be a smooth Fano threefold such that a canonical extension of the tangent bundle is an affine manifold. We show that M is rational homogeneous. [ABSTRACT FROM AUTHOR]

Given an NQC log canonical generalized pair (X , B + M) whose underlying variety X is not necessarily Q -factorial, we show that one may run a (K X + B + M) -MMP with scaling of an ample divisor which terminates, provided that (X , B + M) has a minimal model in a weaker sense or that K X + B + M is not pseudo-effective. We also prove the existence of minimal models of pseudo-effective NQC log canonical generalized pairs under various additional assumptions, for instance when the boundary contains an ample divisor. [ABSTRACT FROM AUTHOR]

In this paper, we define the potential log canonical threshold and prove the ascending chain condition of the set of these thresholds satisfies. We also consider collections of Fano type varieties and study their basic properties including boundedness. [ABSTRACT FROM AUTHOR]

Let X be a smooth, complex Fano 4-fold, and ρ X its Picard number. We show that if ρ X > 12 , then X is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions f : X → Y such that dim f (Exc (f)) = 2 , together with the author's previous work on Fano 4-folds. In particular, given f : X → Y as above, under suitable assumptions we show that S : = f (Exc (f)) is a smooth del Pezzo surface with − K S = (− K Y) | S . [ABSTRACT FROM AUTHOR]

The absolute regularity of a Fano variety, denoted by reg ^ (X) , is the largest dimension of the dual complex of a log Calabi–Yau structure on X. The absolute coregularity is defined to be coreg ^ (X) : = dim X - reg ^ (X) - 1. The coregularity is the complementary dimension of the regularity. We expect that the coregularity of a Fano variety governs, to a large extent, the geometry of X. In this note, we review the history of Fano varieties, give some examples, survey some theorems, introduce the coregularity, and propose several problems regarding this invariant of Fano varieties. [ABSTRACT FROM AUTHOR]

Following Shokurov's idea, we give a simple proof of the ACC conjecture for minimal log discrepancies for surfaces. [ABSTRACT FROM AUTHOR]

In our previous paper [31], we show that all primeℚ-Fano 3-folds X with only 1/2(1, 1, 1)-singularities in certain 5 classes can be embedded as linear sections into bigger dimensionalℚ-Fano varieties called key varieties; each key variety is constructed from data of the Sarkisov link starting from the blow-up at one 1/2(1, 1, 1)-singularity of X. In this paper, we introduce varieties associated with the key varieties which are dual in a certain sense. As an application, we interpret a fundamental part of the Sarkisov link for each X as a linear section of the dual variety. In a natural context describing the variety dual to the key variety of X of genus 5 with one 1/2(1, 1, 1)-singularity, we also characterize a general canonical curve of genus 9 with a g 7 2. [ABSTRACT FROM AUTHOR]

In this article we show that if (X , Δ) is a log canonical compact Kähler threefold such that K X + Δ is nef and the numerical dimension ν (K X + Δ) ≠ 2 , then K X + Δ is semi-ample. [ABSTRACT FROM AUTHOR]

We prove the abundance theorem for arithmetic klt threefold pairs whose closed point have residue characteristic greater than 5. As a consequence, we give a sufficient condition for the asymptotic invariance of plurigenera for certain families of singular surface pairs to hold in mixed characteristic. [ABSTRACT FROM AUTHOR]

The embedded Nash problem for a hypersurface in a smooth algebraic variety is to characterize geometrically the maximal irreducible families of arcs with fixed order of contact along the hypersurface. We show that divisors on minimal models of the pair contribute with such families. We solve the problem for unibranch plane curve germs, in terms of the resolution graph. These are embedded analogs of known results for the classical Nash problem on singular varieties.

We prove several boundedness statements for geometrically integral normal del Pezzo surfaces X over arbitrary fields. We give an explicit sharp bound on the irregularity if X is canonical or regular. In particular, we show that wild canonical del Pezzo surfaces exist only in characteristic $2$ . As an application, we deduce that canonical del Pezzo surfaces form a bounded family over $\mathbb {Z}$ , generalising work of Tanaka. More generally, we prove the BAB conjecture on the boundedness of $\varepsilon $ -klt del Pezzo surfaces over arbitrary fields of characteristic different from $2, 3$ and $5$ .

We prove a decomposition theorem for the nef cone of smooth fiber products over curves, subject to the necessary condition that their Néron–Severi space decomposes. We apply it to describe the nef cone of so-called Schoen varieties, which are the higher-dimensional analogues of the Calabi–Yau threefolds constructed by Schoen. Schoen varieties give rise to Calabi–Yau pairs, and in each dimension at least three, there exist Schoen varieties with nonpolyhedral nef cone. We prove the Kawamata–Morrison–Totaro cone conjecture for the nef cones of Schoen varieties, which generalizes the work by Grassi and Morrison.

For certain quasismooth Calabi–Yau hypersurfaces in weighted projective space, the Berglund-Hübsch-Krawitz (BHK) mirror symmetry construction gives a concrete description of the mirror. We prove that the minimal log discrepancy of the quotient of such a hypersurface by its toric automorphism group is closely related to the weights and degree of the BHK mirror. As an application, we exhibit klt Calabi–Yau varieties with the smallest known minimal log discrepancy. We conjecture that these examples are optimal in every dimension.

We classify finite subgroups G ⊂ PGL 4 (C) such that P 3 is not G-birational to conic bundles and del Pezzo fibrations, and explicitly describe all G-Mori fibre spaces that are G-birational to P 3 for these subgroups. [ABSTRACT FROM AUTHOR]

We present a series of algorithms for computing geometric and representation-theoretic invariants of Calogero–Moser spaces and rational Cherednik algebras associated with complex reflection groups. In particular, we are concerned with Calogero–Moser families (which correspond to the C × -fixed points of the Calogero–Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig’s constructible characters based on a Galois covering of the Calogero–Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a Q -factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity. Making possible these computations was also a source of inspiration for the first author to propose several conjectures about the geometry of Calogero–Moser spaces (cohomology, fixed points, symplectic leaves), often in relation with the representation theory of finite reductive groups. [ABSTRACT FROM AUTHOR]

We study wall-crossing for the Beauville–Mukai system of rank three on a general genus two K3 surface. We show that such a system is related to the Hilbert scheme of ten points on the surface by a sequence of flops, whose exceptional loci can be described as Brill–Noether loci. We also obtain Brill–Noether type results for sheaves in the Beauville–Mukai system. [ABSTRACT FROM AUTHOR]

We show that five of Reid's Fano 3-fold hyperurfaces containing at least one compound Du Val singularity of type c A n have pliability at least two. The two elements of the pliability set are the singular hypersurface itself, and another non-isomorphic Fano hypersurface of the same degree, embedded in the same weighted projective space, but with different compound Du Val singularities. The birational map between them is the composition of two birational links initiated by blowing up two Type I centres on a codimension 4 Fano 3-fold of P 2 × P 2 -type having Picard rank 2. [ABSTRACT FROM AUTHOR]

Let X be a smooth Fano fourfold admitting a conic bundle structure. We show that X is toric if and only if X admits an amplified endomorphism; in this case, X is a rational variety. [ABSTRACT FROM AUTHOR]

Let X be a Fano manifold with Picard number one such that the tangent bundle T X is big. If X admits a rational curve with trivial normal bundle, we show that X is isomorphic to the del Pezzo threefold of degree five. [ABSTRACT FROM AUTHOR]

We prove a classification result for smooth complex Fano fourfolds of Picard number 3 having a prime divisor of Picard number 1, after a characterisation result in arbitrary dimension by Casagrande and Druel [5]. These varieties are obtained by blowing-up a ℙ1-bundle over a smooth Fano variety of Picard number 1 along a codimension 2 subvariety. We study in detail the case of dimension 4, and show that they form 28 families. We compute the main numerical invariants, determine the base locus of the anticanonical system, and study their deformations to give an upper bound to the dimension of the base of the Kuranishi family of a general member. [ABSTRACT FROM AUTHOR]

We prove the termination of flips for pseudo-effective NQC log canonical generalized pairs of dimension 4. As main ingredients, we verify the termination of flips for 3-dimensional NQC log canonical generalized pairs, and show that the termination of flips for pseudo-effective NQC log canonical generalized pairs which admit NQC weak Zariski decompositions follows from the termination of flips in lower dimensions. [ABSTRACT FROM AUTHOR]

This paper proposes the use of F-split and globally F-regular conditions in the pursuit of BAB type results in positive characteristic. The main technical work comes in the form of a detailed study of threefold Mori fibre spaces over positive dimensional bases. As a consequence we prove the main theorem, which reduces birational boundedness for a large class of varieties to the study of prime Fano varieties. [ABSTRACT FROM AUTHOR]

We classify nonrational Fano threefolds X with terminal Gorenstein singularities such that rk Pic (X) = 1 , (- K X) 3 ≥ 8 , and rk Cl (X) ≤ 2 . [ABSTRACT FROM AUTHOR]

We show that, for any fixed weight, there is a natural system of Hodge sheaves, whose Higgs field has no poles, arising from a flat projective family of varieties parametrized by a regular complex base scheme, extending the analogous classical result for smooth projective families due to Griffiths. As an application, based on positivity of direct image sheaves, we establish a criterion for base spaces of rational Gorenstein families to be of general type. A key component of our arguments is centered around the construction of derived categorical objects generalizing relative logarithmic forms for smooth maps and their functorial properties. [ABSTRACT FROM AUTHOR]

A Zariski open subset of an algebraic variety is called a cylinder if it is isomorphic to the direct product of the affine line and an algebraic variety. We consider the existing condition of relative cylinders with respect to a projective dominant morphism of relative dimension two. Since this consideration is essentially a determination of the existence of cylinders in the generic fiber, we study smooth projective surfaces defined over a perfect field from the point of view of cylinders. In previous work, the existing condition of cylinders in smooth minimal del Pezzo surfaces over a field of characteristic zero is known. In this article, we completely determine the existing condition of cylinders in smooth minimal geometrically rational surfaces over a perfect field. Furthermore, we show that for any birational map between smooth projective surfaces over a perfect field, one contains a cylinder if and only if so does the other. [ABSTRACT FROM AUTHOR]

We give a series of examples of log Fano manifolds in any dimension greater than or equal to three by using successive blow-ups along subvarieties in products of two projective spaces. More precisely, we first blow-up a product of two projective spaces along a smooth hypersurface contained in a fiber of one of the two projections and then along the strict transform of a fiber (intersecting the center of the first blow-up) of the other projection. By computing the nef cone and giving an explicit boundary divisor, we show that the resulting variety is always a log Fano manifold. Moreover, the effective cone and the extremal contractions are described. Note that this variety depends on three integral parameters: the dimensions of the two projective spaces and the degree of the hypersurface, which is the center of the first blow-up. Hence, we obtain a series of examples of log Fano manifolds. We also determine which ones are weak Fano. [ABSTRACT FROM AUTHOR]

We show that the Nakai–Moishezon ampleness criterion holds for real line bundles on complete schemes. As applications, we treat the relative Nakai–Moishezon ampleness criterion for real line bundles and the Nakai–Moishezon ampleness criterion for real line bundles on complete algebraic spaces. The main ingredient of this paper is Birkar's characterization of augmented base loci of real divisors on projective schemes. [ABSTRACT FROM AUTHOR]

Let f : X → Y be an algebraic fibre space between normal projective varieties and F be a general fibre of f. We prove an Iitaka-type inequality κ (X , - K X) ≤ κ (F , - K F) + κ (Y , - K Y) under some mild conditions. We also obtain results relating the positivity of - K X and - K Y . [ABSTRACT FROM AUTHOR]

We study infinitesimal deformations of autodual and hyper-holomorphic connections on complex vector bundles on hyper-Kähler manifolds of arbitrary dimension. In particular, we describe the DG Lie algebra controlling this deformation problem. Moreover, we prove associative formality for derived endomorphisms of a holomorphic vector bundle admitting a projectively hyper-holomorphic connection. [ABSTRACT FROM AUTHOR]