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We describe the resolution of singularities of a threefold which has minimal Picard number. We describe the relation between this minimal resolution and an arbitrary resolution of singularities. [ABSTRACT FROM AUTHOR]

Let S be a KLT threefold singularity over an algebraically closed field of positive characteristic p > 5. We prove that its localétale fundamental group is tame and finite. Further, we show that every finite unipotent torsor over a big open subset of S is realized as the restriction of a finite unipotent torsor over S. [ABSTRACT FROM AUTHOR]

A Fano–Enrique's threefold is a 3-dimensional non-Goren stein Fano variety of index 1 with at most canonical singularities. We study the birational geometry of Fano– Enrique's threefold with terminal cyclic quotient singularities. We investigate their rationality and also provide an example of a Fano–Enrique's threefold whose pliability is 9, that is, a Fano–Enrique's threefold birationally equivalent to exactly nine Mori fibre spaces in the Sarkisov category. [ABSTRACT FROM AUTHOR]

We consider Bridgeland stability conditions for threefolds conjectured by Bayer-Macri-Toda in the case of Picard rank 1. We study the differential geometry of numerical walls, characterizing when they are bounded, discussing possible intersections and showing that they are essentially regular. Next, we prove that walls within a certain region of the upper half plane that parameterize geometric stability conditions must always intersect the curve given by the vanishing of the slope function and, for a fixed value of s, have a maximum turning point there. We then use these facts to prove that Gieseker semistability is equivalent to a strong form of asymptotic semistability along a class of paths in the upper half plane, and we show how to find large families of walls. We illustrate how to compute all of the walls and describe the Bridgeland moduli spaces for the Chern character (2,0,-1,0) on complex projective 3-space in a suitable region of the upper half plane. [ABSTRACT FROM AUTHOR]

General hyperplane sections of a Fano threefold Y of index 2 and Picard rank 1 are del Pezzo surfaces, and their Picard group is related to a root system. To the corresponding roots, we associate objects in the Kuznetsov component of Y and investigate their moduli spaces, using the stability condition constructed by Bayer, Lahoz, Macri, and Stellari, and the Abel-Jacobi map. We identify a subvariety of the moduli space isomorphic to Y itself, and as an application we prove a (refined) categorical Torelli theorem for general quartic double solids. [ABSTRACT FROM AUTHOR]

In this paper, we provide the complete classification of ℙ¹-bundles over smooth projective rational surfaces whose neutral component of the automorphism group is maximal. Our results hold over any algebraically closed field of characteristic zero. [ABSTRACT FROM AUTHOR]

We introduce and study extensions of the nef cone of divisors to the full numerical ring of a projective manifold. These are convex cones generated by positive and in one case also semistable vector bundles of arbitrary rank. They share some of the properties of the nef cone. Drezet's log-Chern character motivates us to revisit Bogomolov's inequality, and we find a version for threefolds that involves asymptotic cohomological functions. [ABSTRACT FROM AUTHOR]

We prove the WO-rationality of klt threefolds and the rational chain connectedness of klt Fano threefolds over a perfect field of characteristic p > 5. As a consequence, any klt Fano threefold over a finite field has a rational point. [ABSTRACT FROM AUTHOR]

We prove Grothendieck's conjecture on Resolution of Singularities for quasi-excellent schemes X of dimension three and of arbitrary characteristic. This applies in particular to X = Spec A , A a reduced complete Noetherian local ring of dimension three and to algebraic or arithmetical varieties of dimension three. Similarly, if F is a number field, a complete discretely valued field or more generally the quotient field of any excellent Dedekind domain O , any regular projective surface X / F has a proper and flat model X over O which is everywhere regular. [ABSTRACT FROM AUTHOR]

We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational. [ABSTRACT FROM AUTHOR]

Abstract Smooth threefold flops with irreducible centres are classified by the length invariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katz and Morrison identifies 6 possible partial resolutions of Kleinian singularities that can occur as generic hyperplane sections, and the simultaneous resolutions associated to such a partial resolution produce the universal flop of length l. In this paper we translate these ideas into noncommutative algebra. We introduce the universal flopping algebra of length l from which the universal flop of length l can be recovered by a moduli construction, and we present each of these algebras as the path algebra of a quiver with relations. This explicit realisation can then be used to construct examples of NCCRs associated threefold flops of any length as quiver with relations defined by superpotentials, to recover the matrix factorisation description of the universal flop conjectured by Curto and Morrison, and to realise examples of contraction algebras. [ABSTRACT FROM AUTHOR]

Abstract We study the moduli space of pairs (X , H) consisting of a cubic threefold X and a hyperplane H in P 4. The interest in this moduli comes from two sources: the study of certain weighted hypersurfaces whose middle cohomology admit Hodge structures of K 3 type and, on the other hand, the study of the singularity O 16 (the cone over a cubic surface). In this paper, we give a Hodge theoretic construction of the moduli space of cubic pairs by relating (X , H) to certain "lattice polarized" cubic fourfolds Y. A period map for the pairs (X , H) is then defined using the periods of the cubic fourfolds Y. The main result is that the period map induces an isomorphism between a GIT model for the pairs (X , H) and the Baily–Borel compactification of some locally symmetric domain of type IV. [ABSTRACT FROM AUTHOR]

We classify all projective manifolds with flat holomorphic conformal structure. The Kähler–Einstein case was treated by Kobayashi and Ochiai, there exists a very short list of possible manifolds. In the non-Kähler–Einstein case a classification was only known in small dimensions: by Kobayashi and Ochiai for complex surfaces and by the authors for projective threefolds. This paper completes the general case. [ABSTRACT FROM AUTHOR]

We study the behaviour of Chern numbers of three-dimensional terminal varieties under divisorial contractions. [ABSTRACT FROM AUTHOR]

R.A. Rao–W. van der Kallen showed that the Vaserstein symbol V Γ ( S R 3 ) from the orbit space of unimodular rows of length three over the coordinate ring of the real three sphere S R 3 modulo elementary action to the elementary symplectic Witt group W E ( Γ ( S R 3 ) ) is not injective. Dhvanita R. Rao–Neena Gupta gave an uncountable family of singular real threefolds A α for which the Vaserstein symbol V A α is not injective. In this paper, we give a countable family of smooth real birationally equivalent threefolds A n for which the Vaserstein symbol V A n is not injective. [ABSTRACT FROM AUTHOR]

In this note we mainly consider abelian varieties isogenous to hyperelliptic Jacobians. In the first part we prove that a very general hyperelliptic Jacobian of genus g ≥ 4 is not isogenous to a non-hyperelliptic Jacobian. As a consequence we obtain that the intermediate Jacobian of a very general cubic threefold is not isogenous to a Jacobian. Another corollary tells that the Jacobian of a very general d -gonal curve of genus g ≥ 4 is not isogenous to a different Jacobian. In the second part we consider a closed subvariety Y ⊂ A g of the moduli space of principally polarized varieties of dimension g ≥ 3 . We show that if a very general element of Y is dominated by the Jacobian of a curve C and dim ⁡ Y ≥ 2 g , then C is not hyperelliptic. In particular, if the general element in Y is simple, its Kummer variety does not contain rational curves. Finally we show that a closed subvariety Y ⊂ M g of dimension 2 g − 1 such that the Jacobian of a very general element of Y is dominated by a hyperelliptic Jacobian is contained either in the hyperelliptic or in the trigonal locus. [ABSTRACT FROM AUTHOR]

The Noether–Lefschetz theorem asserts that any curve in a very general surface X in ℙ 3 of degree d ≥ 4 is a restriction of a surface in the ambient space, that is, the Picard number of X is 1. We proved previously that under some conditions, which replace the condition d ≥ 4 , a very general surface in a simplicial toric threefold ℙ Σ (with orbifold singularities) has the same Picard number as ℙ Σ . Here we define the Noether–Lefschetz loci of quasi-smooth surfaces in ℙ Σ in a linear system of a Cartier ample divisor with respect to a ( − 1 ) -regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether–Lefschetz loci which contain a line, defined as a rational curve which is minimal in a suitable sense. [ABSTRACT FROM AUTHOR]

Kishimoto raised the problem to classify all compactifications of contractible affine 3-folds into smooth Fano 3-folds with second Betti number two and classified such compactifications whose log canonical divisors are not nef. In this paper, we show that there are 14 deformation equivalence classes of smooth Fano 3-folds which can admit structures of such compactifications whose log canonical divisors are trivial. We also construct an example of such compactifications with trivial log canonical divisors for each of all the 14 classes. [ABSTRACT FROM AUTHOR]

Abstract: We present a list of Calabi‐Yau threefolds known to us, and with holonomy groups that are precisely S U ( 3 ), rather than a subgroup, with small Hodge numbers, which we understand to be those manifolds with height ( h 1 , 1 + h 2 , 1 ) ≤ 24. With the completion of a project to compute the Hodge numbers of free quotients of complete intersection Calabi‐Yau threefolds, most of which were computed in Refs. [ ] and the remainder in Ref. [ ], many new points have been added to the tip of the Hodge plot, updating the reviews by Davies and Candelas in Refs. [ ]. In view of this and other recent constructions of Calabi‐Yau threefolds with small height, we have produced an updated list. [ABSTRACT FROM AUTHOR]

We give a formula for Hodge numbers of double coverings with h 2 , 0 = 0 over compact Kähler manifolds. As an application, we consider Calabi–Yau double coverings and calculate their Hodge numbers. In this way, we find several pairs ( h 1 , 1 , h 1 , 2 ) of Hodge numbers of Calabi–Yau threefolds that do not come from toric setting. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove the original Bogomolov–Gieseker type inequality conjecture for P 1 × S , P 2 × C and P 1 × P 1 × C, where S is an Abelian surface and C is an elliptic curve. In particular, there exist Bridgeland stability conditions on these threefolds. [ABSTRACT FROM AUTHOR]

Brown and Schnetz found that the number of points over F p of a graph hypersurface is often related to the coefficients of a modular form. We set some of the reduction techniques used to discover such relations in a general geometric context. We also prove the relation for two examples of modular forms of weight 3 and two of weight 4, refine the statement and suggest a method of proving it for three more of weight 4, and use one of the proved examples to construct two new rigid Calabi–Yau threefolds that realize Hecke eigenforms of weight 4 (one provably and one conjecturally). [ABSTRACT FROM AUTHOR]

Deployable structures have the advantage of being able to change their size and morphology significantly with minimal mobility. Yet, there are very limited numbers of deployable robots. This paper proposes a transformable robot by applying a threefold-symmetric Bricard linkage as the body structure. The geometrics of the robot are investigated to set up relationships between its height/foot span and the joint angles of the linkage. The locomotion gaits of the robot can be realized through the deploying and folding motions of the Bricard linkage. From this, the corresponding gait controller is designed. Experimental results show that the robot can move in an arbitrary direction and follow a given path. Moreover, it is capable of moving through limited space easily by changing its configuration from folded to deployed, and vice versa. [ABSTRACT FROM AUTHOR]

Abstract: We show that Rost nilpotence holds for a geometrically integral threefold X over a field k of characteristic 0 if and only if α k ( X ) * ∘ N ( CH 0 ( X k ( X ) ) ) = 0 for some integer N > 0 for all correspondences α of degree 0 which vanish over some field extension of k. As a corollary we get the Rost nilpotence property for three‐dimensional smooth projective geometrically integral schemes over a field of characteristic zero, which are birationally isomorphic to a toric model. [ABSTRACT FROM AUTHOR]

We study duality spectral sequences for Weierstraß fibrations. Using these spectral sequences, we show that on a K -trivial Weierstraß threefold over a K -numerically trivial surface, any line bundle of nonzero fiber degree is taken by a Fourier–Mukai transform to a slope stable locally free sheaf. [ABSTRACT FROM AUTHOR]

We observe that an interesting method to produce non-complete intersection subvarieties, the generalized complete intersections from L. Anderson and coworkers, can be understood and made explicit by using standard Cech cohomology machinery. We include a worked example of a generalized complete intersection Calabi–Yau threefold. [ABSTRACT FROM AUTHOR]

Using the techniques introduced by Corvaja and Zannier in 2008 we solve the non-split case of the geometric Lang-Vojta Conjecture for affine surfaces isomorphic to the complement of a conic and two lines in the projective plane. In this situation we deal with sections of an affine threefold fibered over a curve, whose boundary, in the natural projective completion, is a quartic bundle over the base whose fibers have three irreducible components. We prove that the image of each section has bounded degree in terms of the Euler characteristic of the base curve. [ABSTRACT FROM AUTHOR]

Any Calabi-Yau three-fold X with infinite fundamental group admits an étale Galois covering either by an abelian three-fold or by the product of a K3 surface and an elliptic curve.Wecall X of type A in the former case and of type K in the latter case. In this article, we provide the full classification of Calabi-Yau three-folds of type K, based on Oguiso and Sakurai's work [25]. Together with a refinement of their result on Calabi-Yau threefolds of type A, we finally complete the classification of Calabi-Yau three-folds with infinite fundamental group. [ABSTRACT FROM AUTHOR]

Weinvestigate the moduli space of instanton bundles over the quintic del Pezzo threefold for small values of their charge. We prove a new bound for their splitting type on a generic line and an analogue of the Grauert-Mülich theorem for minimal instantons. Moreover, in the first two non-trivial cases, we construct SL2-equivariant embeddings into ℙ5 and into a relative Grassmannian. These results are used to prove the existence of a unique SL2-equivariant instanton of charge 2 and to provide examples of singular theta-characteristics associated with instantons of charge 3. [ABSTRACT FROM AUTHOR]

We investigate some coboundary map associated to a 3-fold terminal singularity which is important in the study of deformations of singular 3-folds. We prove that this map vanishes only for quotient singularities and an A1,2/4-singularity, that is, a terminal singularity analytically isomorphic to a Z4-quotient of the singularity (x² +y² + z³ + u² = 0). As an application, we prove that a Q-Fano 3-fold with terminal singularities can be deformed to one with only quotient singularities and A1,2/4-singularities. We also treat the Q-smoothability problem on Q-Calabi--Yau 3-folds. [ABSTRACT FROM AUTHOR]

We construct Calabi–Yau threefolds with relatively small Hodge numbers h 1 , 1 's by smoothing normal crossing varieties, which are obtained from Fano threefolds. We consider over 300 configurations and compute Hodge numbers of Calabi–Yau threefolds. Many of those Hodge pairs ( h 1 , 1 , h 1 , 2 ) do not overlap with those of Calabi–Yau threefolds constructed in the toric setting. [ABSTRACT FROM AUTHOR]

We utilize machine learning to study the string landscape. Deep data dives and conjecture generation are proposed as useful frameworks for utilizing machine learning in the landscape, and examples of each are presented. A decision tree accurately predicts the number of weak Fano toric threefolds arising from reflexive polytopes, each of which determines a smooth F-theory compactification, and linear regression generates a previously proven conjecture for the gauge group rank in an ensemble of $$ \frac{4}{3}\times 2.96\times {10}^{755} $$ F-theory compactifications. Logistic regression generates a new conjecture for when E arises in the large ensemble of F-theory compactifications, which is then rigorously proven. This result may be relevant for the appearance of visible sectors in the ensemble. Through conjecture generation, machine learning is useful not only for numerics, but also for rigorous results. [ABSTRACT FROM AUTHOR]

We study ample divisors X with only rational singularities on abelian varieties that decompose into a sum of two lower dimensional subvarieties, X = V + W. For instance, we prove an optimal lower bound on the degree of the addition map V × W → X and show that the minimum can only be achieved if X is a theta divisor. Conjecturally, the latter happens only on Jacobians of curves and intermediate Jacobians of cubic threefolds. As an application, we prove that nondegenerate generic vanishing subschemes of indecomposable principally polarized abelian varieties are automatically reduced and irreducible, have the expected geometric genus and property (P) with respect to their theta duals. [ABSTRACT FROM AUTHOR]

We study the geometric description of BPS states in supersymmetric theories with eight supercharges in terms of geodesic networks on suitable spectral curves. We lift and extend several constructions of Gaiotto-Moore-Neitzke from gauge theory to local Calabi-Yau threefolds and related models. The differential is multi-valued on the covering curve and features a new type of logarithmic singularity in order to account for D0-branes and non-compact D4-branes, respectively. We describe local rules for the three-way junctions of BPS trajectories relative to a particular framing of the curve. We reproduce BPS quivers of local geometries and illustrate the wall-crossing of finite-mass bound states in several new examples. We describe first steps toward understanding the spectrum of framed BPS states in terms of such 'exponential networks'. [ABSTRACT FROM AUTHOR]

The aim of this paper is to complete the classification of all Calabi–Yau threefolds which are constructed as the quotient of a smooth Schoen threefold X = B 1 × P 1 B 2 (fiber product over P 1 of two relatively minimal rational elliptic surfaces B 1 and B 2 with section) under a finite group action acting freely on the Schoen threefold X . The abelian group actions on smooth Schoen threefolds which induce cyclic group actions on the base curve P 1 were studied by Bouchard and Donagi (2008), and all such actions were listed. We consider the actions on the Schoen threefold by finite groups G whose elements are given as a product τ 1 × τ 2 of two automorphisms τ 1 and τ 2 of the rational elliptic surfaces B 1 and B 2 with section. In this paper, we use the classification of automorphism groups of rational elliptic surfaces with section given in Karayayla (2012) and Karayayla (2014) to generalize the results of Bouchard and Donagi to answer the question whether finite and freely acting group actions on Schoen threefolds which induce non-cyclic group actions on the base curve P 1 exist or not. Despite the existence of group actions on rational elliptic surfaces which induce non-cyclic (even non-abelian) group actions on P 1 , it is shown in this paper that none of those actions can be lifted to free actions on a Schoen threefold. The main result is that there is no finite group action on a Schoen threefold X which acts freely on X and which induces a non-cyclic group action on the base curve P 1 . This result shows that the list given in Bouchard and Donagi (2008) is a complete list of non-simply connected Calabi–Yau threefolds constructed as the quotient of a smooth Schoen threefold by a finite group action. [ABSTRACT FROM AUTHOR]

We show that a proper algebraic n-dimensional scheme Y admits non-trivial vector bundles of rank n, even if 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. [ABSTRACT FROM AUTHOR]

LetXbe the product of two projective spaces and consider the general CICY threefoldYinXwith configuration matrixA. We prove the finiteness part of the analogue of the Clemens’ conjecture for such a CICY in low bidegrees. More precisely, we prove that the number of smooth rational curves onYwith low bidegree and with nondegenerate birational projection is at most finite (even in cases in which positive dimensional families of degenerate rational curves are known). [ABSTRACT FROM PUBLISHER]

By Jahnke-Peternell-Radloff and Takeuchi, almost Fano threefolds with del Pezzo fibrations were classified. Among them, there exist 10 classes such that the existence of members of these was not proved. In this paper, we construct such examples belonging to each of 10 classes. [ABSTRACT FROM AUTHOR]

We give a counterexample to the generalized Bogomolov-Gieseker inequality for threefolds conjectured by Bayer, Macrì, and Toda using the blow up of a point in three-dimensional projective space. [ABSTRACT FROM AUTHOR]

This work deals with the study of embeddings of toric Calabi–Yau fourfolds which are complex cones over the smooth Fano threefolds. In particular, we focus on finding various embeddings of Fano threefolds inside other Fano threefolds and study the partial resolution of the latter in hope to find new toric dualities. We find many diagrams possible for many of these Fano threefolds, but unfortunately, none of them are consistent quiver theories. We also obtain a quiver Chern–Simons theory which matches a theory known to the literature, thus providing an alternate method of obtaining it. [ABSTRACT FROM AUTHOR]

A threefold isogenous to a product of curves X is a quotient of a product of three compact Riemann surfaces of genus at least two by the free action of a finite group. In this paper we study these threefolds under the assumption that the group acts diagonally on the product. We show that the classification of these threefolds is a finite problem, present an algorithm to classify them for a fixed value of χ ( O X ) and explain a method to determine their Hodge numbers. Running an implementation of the algorithm we achieve the full classification of threefolds isogenous to a product of curves with χ ( O X ) = − 1 , under the assumption that the group acts faithfully on each factor. [ABSTRACT FROM AUTHOR]

The article discusses the study about automorphisms of singular cubic threefolds X and the Cremona group. Topics discussed include the following notations for groups in studying this paper such as C n and as the cyclic group and S n and as the symmetric group, the G-equivalent birational map X and a proof of assertion regarding the automorphism of X.