In this paper, we prove the birational rigidity of Fano-Mori fibre spaces $\pi \colon V\to S$, every fibre of which is a Fano complete intersection of index 1 and codimension $k\geqslant 3$ in the projective space ${\mathbb P}^{M+k}$ for M sufficiently high, satisfying certain natural conditions of general position, in the assumption that the fibre space $V/S$ is sufficiently twisted over the base. The dimension of the base S is bounded from above by a constant, depending only on the dimension M of the fibre (as the dimension of the fibre M grows, this constant grows as $\frac 12 M^2$).

Sextic double solids, double covers of $\mathbb P^3$ branched along a sextic surface, are the lowest degree Gorenstein terminal Fano 3-folds, hence are expected to behave very rigidly in terms of birational geometry. Smooth sextic double solids, and those which are $\mathbb Q$-factorial with ordinary double points, are known to be birationally rigid. In this paper, we study sextic double solids with an isolated compound $A_n$ singularity. We prove a sharp bound $n \leq 8$, describe models for each n explicitly, and prove that sextic double solids with $n> 3$ are birationally nonrigid.

In this paper, we prove the nonvanishing and some special cases of the abundance for log canonical threefold pairs over an algebraically closed field k of characteristic $p> 3$. More precisely, we prove that if $(X,B)$ be a projective log canonical threefold pair over k and $K_{X}+B$ is pseudo-effective, then $\kappa (K_{X}+B)\geq 0$, and if $K_{X}+B$ is nef and $\kappa (K_{X}+B)\geq 1$, then $K_{X}+B$ is semi-ample.

As applications, we show that the log canonical rings of projective log canonical threefold pairs over k are finitely generated and the abundance holds when the nef dimension $n(K_{X}+B)\leq 2$ or when the Albanese map $a_{X}:X\to \mathrm {Alb}(X)$ is nontrivial. Moreover, we prove that the abundance for klt threefold pairs over k implies the abundance for log canonical threefold pairs over k.

For a terminal weak ${\mathbb {Q}}$-Fano threefold X, we show that the mth anti-canonical map defined by $|-mK_X|$ is birational for all $m\geq 59$.

We give conditions for a uniruled variety of dimension at least 2 to be nonsolid. This study provides further evidence to a conjecture by Abban and Okada on the solidity of Fano 3-folds. To complement our results we write explicit birational links from Fano 3-folds of high codimension embedded in weighted projective spaces.

We give an explicit characterization on the singularities of exceptional pairs in any dimension. In particular, we show that any exceptional Fano surface is $\frac {1}{42}$ -lc. As corollaries, we show that any $\mathbb R$ -complementary surface X has an n-complement for some integer $n\leq 192\cdot 84^{128\cdot 42^5}\approx 10^{10^{10.5}}$ , and Tian’s alpha invariant for any surface is $\leq 3\sqrt {2}\cdot 84^{64\cdot 42^5}\approx 10^{10^{10.2}}$ . Although the latter two values are expected to be far from being optimal, they are the first explicit upper bounds of these two algebraic invariants for surfaces.

Let V be a smooth quasi-projective complex surface such that the first three logarithmic plurigenera $\overline P_1(V)$ , $\overline P_2(V)$ and $\overline P_3(V)$ are equal to 1 and the logarithmic irregularity $\overline q(V)$ is equal to $2$ . We prove that the quasi-Albanese morphism $a_V\colon V\to A(V)$ is birational and there exists a finite set S such that $a_V$ is proper over $A(V)\setminus S$ , thus giving a sharp effective version of a classical result of Iitaka [12].

The pentagram map, introduced by Schwartz [The pentagram map. Exp. Math. 1(1) (1992), 71–81], is a dynamical system on the moduli space of polygons in the projective plane. Its real and complex dynamics have been explored in detail. We study the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2. We prove that the pentagram map on twisted polygons is a discrete integrable system, in the sense of algebraic complete integrability: the pentagram map is birational to a self-map of a family of abelian varieties. This generalizes Soloviev’s proof of complex integrability [F. Soloviev. Integrability of the pentagram map. Duke Math. J. 162(15) (2013), 2815–2853]. In the course of the proof, we construct the moduli space of twisted n-gons, derive formulas for the pentagram map, and calculate the Lax representation by characteristic-independent methods.

A famous problem in birational geometry is to determine when the birational automorphism group of a Fano variety is finite. The Noether–Fano method has been the main approach to this problem. The purpose of this paper is to give a new approach to the problem by showing that in every positive characteristic, there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphism group. To do this, we prove that these varieties admit ample and birationally equivariant line bundles. Our result applies the differential forms that Kollár produces on $p$-cyclic covers in characteristic $p > 0$.

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.

Varieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees $1$ and $2$ over the projective line with smooth total space satisfying the so-called $K^2$ -condition are birationally rigid: their Mori fiber space structure is unique. This implies that they are not birational to any Fano varieties, conic bundles, or other del Pezzo fibrations. In particular, they are irrational. The families of del Pezzo fibrations with smooth total space of degree $2$ are rather special, as for most families a general del Pezzo fibration has the simplest orbifold singularities. We prove that orbifold del Pezzo fibrations of degree $2$ over the projective line satisfying explicit generality conditions as well as a generalized $K^2$ -condition are birationally rigid.

We study derived categories of Gorenstein varieties $X$ and $X^+$ connected by a flop. We assume that the flopping contractions $f\colon X\to Y$, $f^+ \colon X^+ \to Y$ have fibers of dimension bounded by one and $Y$ has canonical hypersurface singularities of multiplicity two. We consider the fiber product $W=X\times _YX^+$ with projections $p\colon W\to X$, $p^+\colon W\to X^+$ and prove that the flop functors $F = Rp^+_*Lp^* \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^b(X^+)$, $F^+= Rp_*L{p^+}^* \colon {\mathcal {D}}^b(X^+) \to {\mathcal {D}}^b(X)$ are equivalences, inverse to those constructed by Van den Bergh. The composite $F^+ \circ F \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^b(X)$ is a non-trivial auto-equivalence. When variety $Y$ is affine, we present $F^+ \circ F$ as the spherical cotwist of a spherical couple $(\Psi ^*,\Psi )$ which involves a spherical functor $\Psi$ constructed by deriving the inclusion of the null category $\mathscr {A}_f$ of sheaves ${\mathcal {F}} \in \mathop {{\rm Coh}}\nolimits (X)$ with $Rf_*({\mathcal {F}} )=0$ into $\mathop {{\rm Coh}}\nolimits (X)$. We construct a spherical pair (${\mathcal {D}}^b(X)$, ${\mathcal {D}}^b(X^+)$) in the quotient ${\mathcal {D}}^b(W) /{\mathcal {K}}^b$, where ${\mathcal {K}}^b$ is the common kernel of the derived push-forwards for the projections to $X$ and $X^+$, thus implementing in geometric terms a schober for the flop. A technical innovation of the paper is the $L^1f^*f_*$ vanishing for Van den Bergh's projective generator. We construct a projective generator in the null category and prove that its endomorphism algebra is the contraction algebra.

In this paper, we prove a decomposition result for the Chow groups of projectivizations of coherent sheaves of homological dimension $\le 1$ . In this process, we establish the decomposition of Chow groups for the cases of the Cayley trick and standard flips. Moreover, we apply these results to study the Chow groups of symmetric powers of curves, nested Hilbert schemes of surfaces, and the varieties resolving Voisin maps for cubic fourfolds.

In this article, we describe the embeddings of the Heisenberg group into the Cremona group.

A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

We compactify and regularise the space of initial values of a planar map with a quartic invariant and use this construction to prove its integrability in the sense of algebraic entropy. The system has certain unusual properties, including a sequence of points of indeterminacy in $\mathbb {P}^{1}\!\times \mathbb {P}^{1}$ . These indeterminacy points lie on a singular fibre of the mapping to a corresponding QRT system and provide the existence of a one-parameter family of special solutions.

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index $e$, then the degree of irrationality of a very general complex Fano hypersurface of index $e$ and dimension n is bounded from below by a constant times $\sqrt{n}$. To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic $p$ argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.

For infinitely many d, Hassett showed that special cubic fourfolds of discriminant d are related to polarised K3 surfaces of degree d via their Hodge structures. For half of the d, each associated K3 surface (S, L) canonically yields another one, (Sτ, Lτ). We prove that Sτ is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3, L, d/6). We also explain for which d the Hilbert schemes Hilbn (S) and Hilbn (Sτ) are birational.

We discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky.

We prove that every birationally superrigid Fano variety whose alpha invariant is greater than (respectively no smaller than) $\frac{1}{2}$ is K-stable (respectively K-semistable). We also prove that the alpha invariant of a birationally superrigid Fano variety of dimension $n$ is at least $1/(n+1)$ (under mild assumptions) and that the moduli space (if it exists) of birationally superrigid Fano varieties is separated.