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In this paper, we study (i) -curves with i ∈ { − 1 , 0 , 1 } in the blown-up projective space P r in general points. The notion of (− 1) -curves was analyzed in the early days of mirror symmetry by Kontsevich, with the motivation of counting curves on a Calabi–Yau threefold. In dimension two, Nagata studied planar (− 1) -curves in order to construct a counterexample to Hilbert's 14th problem. We introduce the notion of classes of (0) - and (1) -curves in P r with s points blown up, and we prove that their number is finite if and only if the space is a Mori Dream Space. We further introduce a bilinear form on a space of curves and a unique symmetric Weyl-invariant class, F (which we will refer to as the anticanonical curve class). For Mori Dream Spaces, we prove that (− 1) -curves can be defined arithmetically by the linear and quadratic invariants determined by the bilinear form. Moreover, (0) - and (1) -Weyl lines give the extremal rays for the cone of movable curves in P r with r + 3 points blown up. As an application, we use the technique of movable curves to reprove that if F 2 ≤ 0 then Y is not a Mori Dream Space, and we propose to apply this technique to other spaces. [ABSTRACT FROM AUTHOR]

We study multihomogeneous spaces corresponding to ℤ n -graded algebras over an algebraically closed field of characteristic 0 and their relation with the description of T -varieties. [ABSTRACT FROM AUTHOR]

We describe the automorphism groups of smooth Fano threefolds of rank 2 and degree 28 in the cases where they are finite. [ABSTRACT FROM AUTHOR]

This paper investigates isomorphisms of Bergman fans of matroids respecting different fan structures, which we regard as matroid analogs of birational maps. We show that isomorphisms respecting the fine fan structure are induced by matroid isomorphisms.We introduce Cremona automorphisms of the coarse structure of Bergman fans, which are not induced by matroid automorphisms. We show that the automorphism group of the coarse fan structure is generated by matroid automorphisms and Cremona maps in the case of rank \(3\) matroids which are not parallel connections and for modularly complemented matroids.Mathematics Subject Classifications: 14T20, 52B40, 14E07Keywords: Matroids, birational geometry

We prove that the moduli space of double covers ramified at two points R g , 2 is uniruled for 3 ≤ g ≤ 6 and of general type for g ≥ 16 . Furthermore, we consider Prym-canonical divisorial strata in the moduli space C n R ¯ g parametrizing n-pointed Prym curves, and we compute their classes in Pic Q ( C n R ¯ g) . [ABSTRACT FROM AUTHOR]

We study compactifications of the moduli space of unordered points in the plane via variation of GIT-quotients of their corresponding Hilbert scheme. Our VGIT considers linearizations outside the ample cone and within the movable cone. For that purpose, we use the description of the Hilbert scheme as a Mori dream space, and the moduli interpretation of its birational models via Bridgeland stability. We determine the GIT walls associated with curvilinear zero-dimensional schemes, collinear points, and schemes supported on a smooth conic. For seven points, we study a compactification associated with an extremal ray of the movable cone, where stability behaves very differently from the Chow quotient. Lastly, a complete description for five points is given. [ABSTRACT FROM AUTHOR]

The aim of this paper is to classify codimension 1 foliations F$\operatorname{\mathcal {F}}$ with canonical singularities and ν(KF)<3$\nu (K_{\operatorname{\mathcal {F}}}) < 3$ on threefolds of general type. I prove a classification result for foliations satisfying these conditions and having nontrivial algebraic part. We also describe purely transcendental foliations F$\operatorname{\mathcal {F}}$ with the canonical class KF$K_{\operatorname{\mathcal {F}}}$ being not big on manifolds of general type in any dimension, assuming that F$\operatorname{\mathcal {F}}$ is nonsingular in codimension 2. [ABSTRACT FROM AUTHOR]

In this paper we study geometric realizations of some classic birational maps, such as inversion maps, special Cremona transformations, special birational transformations of type (2, 1), by considering C ∗ -actions on rational homogeneous spaces and their subvarieties. [ABSTRACT FROM AUTHOR]

Smooth, complex Fano varieties are a very natural class of projective varieties, and have been intensively studied both classically and in recent times, due to their significance in the framework of the Minimal Model Program. The Lefschetz defect is an invariant of smooth Fano varieties that has been recently introduced in [7]; it is related to the Picard number ρ X of X, and to the Picard number of prime divisors in X. This paper is a survey on this new invariant and its properties: we explain the definition of the Lefschetz defect δ X and its origin, the known results, and several examples, especially in dimensions 3 and 4; we do not give new theoretical results. In particular, we explain how the results on the Lefschetz defect allow to recover the classification of Fano 3-folds with ρ X ≥ 5 , due to Mori and Mukai. We also review the known examples of Fano 4-folds with ρ X ≥ 6 , that are remarkably few: apart products and toric examples, there are only 6 known families, with ρ X ∈ { 6 , 7 , 8 , 9 } . [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]

In his work on arithmetic Fujita approximation, Huayi Chen introduces the notion of an approximable graded algebra, which he uses to prove a Fujita-type theorem in the arithmetic setting, and asks if any such algebra is the graded ring of a big line bundle on a projective variety. This was proved to be false by the author in a previous work. A subsequent paper showed that, whilst the approximable algebra is not necessarily a subalgebra of the algebra of graded sections of a big line bundle, it is a full-dimensional subalgebra of the algebra of sections of an infinite Weil divisor. This paper also proved that over C, if an infinite Weil divisor has a finite numerical class, then its section ring is approximable, and the same is true for any full-dimensional subalgebra. This note proves that the converse is true: any approximable algebra over C is associated to an infinite Weil divisor whose numerical class converges. [ABSTRACT FROM AUTHOR]

We classify the maximal algebraic subgroups of Bir(C x ℙ¹), when C is a smooth CM projective curve of positive genus. [ABSTRACT FROM AUTHOR]

We consider weak Fano manifolds with small contractions obtained by blowing up successively curves and subvarieties of codimension 2 in the product of a projective space and a projective line. We give a classification result for a special case. We determine the nef cones and describe the extremal contractions for related smooth projective varieties. [ABSTRACT FROM AUTHOR]

Unprojection is a theory due to Reid which constructs more complicated rings starting from simpler data. In the present work, we develop a new method of unprojection. Starting from a codimension 3 ideal defined by the Pfaffians of a 5 × 5 skewsymmetric matrix, we describe a number of unprojection formats that construct Gorenstein rings of codimension 6. We give two applications that construct families of Fano 3-folds embedded anticanonically in codimension 6 that existing methods do not realize. [ABSTRACT FROM AUTHOR]

We rework the Mori–Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians. [ABSTRACT FROM AUTHOR]

We prove that any nef $b$ -divisor class on a projective variety defined over an algebraically closed field of characteristic zero is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef $b$ -divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony. We show that any big and basepoint-free curve class is a power of a nef $b$ -divisor, and relate this statement to the Zariski decomposition of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces contained in the space of $b$ -divisors which were defined in our previous work. [ABSTRACT FROM AUTHOR]

Let O be a nilpotent orbit of a complex semisimple Lie algebra g and let π : X → O ¯ be the finite covering associated with the universal covering of O. In the previous article [14] we have explicitly constructed a Q -factorial terminalization X ˜ of X when g is classical. In this article we count how many non-isomorphic Q -factorial terminalizations X has. We construct the universal Poisson deformation of X ˜ over H 2 (X ˜ , C) and look at the action of the Weyl group W (X) on H 2 (X ˜ , C). The main result is an explicit geometric description of W (X). [ABSTRACT FROM AUTHOR]

We study the existence and properties of birationally equivalent models for elliptically fibered varieties. In particular these have either the structure of Mori fiber spaces or, assuming some standard conjectures, minimal models with a Zariski decomposition compatible with the elliptic fibration. We prove relations between the birational invariants of the elliptically fibered variety, the base of the fibration and of its Jacobian. [ABSTRACT FROM AUTHOR]

We show that, for any nonsingular projective 4-fold V of general type with geometric genus p g ≥ 2 , the pluricanonical map φ 33 is birational onto the image and the canonical volume V o l (V) has the lower bound 1 480 , which improves a previous theorem by Chen and Chen. [ABSTRACT FROM AUTHOR]

Combining theorems of Voisin and Marian, Shen, Yin and Zhao, we compute the dimensions of the orbits under rational equivalence in the Mukai system of rank two and genus two. We produce several examples of algebraically coisotropic and constant cycle subvarieties. [ABSTRACT FROM AUTHOR]

This paper discusses a theorem in birational geometry that J.L. Coolidge attributed to Alfred Clebsch. The background is reconstructed from letters Felix Klein exchanged with Max Noether in 1894, when Noether was completing work on a lengthy report with Alexander Brill on the history of algebraic functions. Noether was deeply troubled to learn that Klein had informed him back in 1869 about relevant results that Clebsch and Leopold Kronecker had discussed in Berlin. These exchanges with Klein led to revisions in the Brill-Noether report, made in part to ensure Noether's own priority rights and larger intellectual legacy. [ABSTRACT FROM AUTHOR]

We develop a generalization of the Q -construction of the first author, Diemer, and the third author for Grassmann flips. This generalization provides a canonical idempotent kernel on the derived category of the associated global quotient stack. The idempotent kernel, after restriction, induces a semi-orthogonal decomposition which compares the flipped varieties. Furthermore its image, after restriction to the geometric invariant theory semistable locus, "opens" a canonical "window" in the derived category of the quotient stack. We check this window coincides with the set of representations used by Kapranov to form a full exceptional collection on Grassmannians. [ABSTRACT FROM AUTHOR]

In this note we introduce generalised pairs from the perspective of the evolution of the notion of space in birational algebraic geometry. We describe some applications of generalised pairs in recent years and then mention a few open problems. [ABSTRACT FROM AUTHOR]

The Separatrix Theorem of C. Camacho and P. Sad says that there exists at least one invariant curve (separatrix) passing through the singularity of a germ of holomorphic foliation on complex surface, when the surface underlying the foliation is smooth or when it is singular and the dual graph of resolution surface singularity is a tree. Under some assumptions, we obtain existence of separatrix even when the resolution dual graph of the surface singular point is not a tree. It will be necessary to require an extra condition of the foliation, namely, absence of saddle-node in its reduction of singularities. [ABSTRACT FROM AUTHOR]

Given a quasi-projective 3-fold X with only Gorenstein terminal singularities, we prove that the flop functors beginning at X satisfy higher degree braid relations, with the combinatorics controlled by a real hyperplane arrangement H. This leads to a general theory, incorporating known special cases with degree 3 braid relations, in which we show that higher degree relations can occur even for two smooth rational curves meeting at a point. This theory yields an action of the fundamental group of the complexified complement, π1(ℂn\Hℂ), on the derived category of X, for any such 3-fold that admits individually floppable curves. We also construct such an action in the more general case where individual curves may flop analytically, but not algebraically, and furthermore we lift the action to a form of affine pure braid group under the additional assumption that X is ℚ-factorial. Along the way, we produce two new types of derived autoequivalences. One uses commutative deformations of the scheme-theoretic fibre of a flopping contraction, and the other uses noncommutative deformations of the fibre with reduced scheme structure, generalising constructions of Toda and the authors [T07, DW1] which considered only the case when the flopping locus is irreducible. For type A flops of irreducible curves, we show that the two autoequivalences are related, but in other cases they are very different, with the noncommutative twist being linked to birational geometry via the Bridgeland--Chen [B02, C02] flop-flop functor. [ABSTRACT FROM AUTHOR]

Abstract Given a contraction of a variety X to a base Y , we enhance the locus in Y over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings D , under mild assumptions which hold in the case of (1) crepant partial resolutions admitting a tilting bundle with trivial summand, and (2) all contractions with fibre dimension at most one. In all cases, this produces a global invariant. In the crepant setting, we then apply this to study derived autoequivalences of X. We work generally, dropping many of the usual restrictions, and so both extend and unify existing approaches. In full generality we construct a new endofunctor of the derived category of X by twisting over D , and then, under appropriate restrictions on singularities, give conditions for when it is an autoequivalence. We show that these conditions hold automatically when the non-isomorphism locus in Y has codimension 3 or more, which covers determinantal flops, and we also control the conditions when the non-isomorphism locus has codimension 2, which covers 3-fold divisor-to-curve contractions. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the moduli space of stable pairs (or sheaves) on del Pezzo surface S is isomorphic to the Maruyama type transformation of a projective bundle of a quiver representation space. [ABSTRACT FROM AUTHOR]