In this paper, we study the positivity property of the tangent bundle $T_X$ of a Fano threefold X with Picard number $2$. We determine the bigness of the tangent bundle of the whole $36$ deformation types. Our result shows that $T_X$ is big if and only if $(-K_X)^3\ge 34$. As a corollary, we prove that the tangent bundle is not big when X has a standard conic bundle structure with non-empty discriminant. Our main methods are to produce irreducible effective divisors on ${\mathbb {P}}(T_X)$ constructed from the total dual VMRT associated to a family of rational curves. Additionally, we present some criteria to determine the bigness of $T_X$.

Fujino gave a proof for the semi-ampleness of the moduli part in the canonical bundle formula in the case when the general fibers are K3 surfaces or abelian varieties. We show a similar statement when the general fibers are primitive symplectic varieties. This answers a question of Fujino raised in the same article. Moreover, using the structure theory of varieties with trivial first Chern class, we reduce the question of semi-ampleness in the case of families of K-trivial varieties to a question when the general fibers satisfy a slightly weaker Calabi–Yau condition.

We show the properness of the moduli stack of stable surfaces over $\mathbb{Z}\left[ {1/30} \right]$, assuming the locally-stable reduction conjecture for stable surfaces. This relies on a local Kawamata–Viehweg vanishing theorem for 3-dimensional log canonical singularities at closed point of characteristic $p \ne 2,3$ and $5$, which are not log canonical centres.

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 $5$.

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.

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.

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 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 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 $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-({K_{X}} + B)$ nef over S. A conjecture, known as the Shokurov–Kollár connectedness principle, predicts that $f^{-1} (s) \cap \operatorname {\mathrm {Nklt}}(X,B)$ has at most two connected components, where $s \in S$ is an arbitrary schematic point and $\operatorname {\mathrm {Nklt}}(X,B)$ denotes the non-klt locus of $(X,B)$ . In this work, we prove this conjecture, characterizing those cases in which $\operatorname {\mathrm {Nklt}}(X,B)$ fails to be connected, and we extend these same results also to the category of generalized pairs. Finally, we apply these results and the techniques to the study of the dual complex for generalized log Calabi–Yau pairs, generalizing results of Kollár–Xu [Invent. Math. 205 (2016), 527–557] and Nakamura [Int. Math. Res. Not. IMRN 13 (2021), 9802–9833].

Let $(X\ni x,B)$ be an lc surface germ. If $X\ni x$ is klt, we show that there exists a divisor computing the minimal log discrepancy of $(X\ni x,B)$ that is a Kollár component of $X\ni x$ . If $B\not=0$ or $X\ni x$ is not Du Val, we show that any divisor computing the minimal log discrepancy of $(X\ni x,B)$ is a potential lc place of $X\ni x$ . This extends a result of Blum and Kawakita who independently showed that any divisor computing the minimal log discrepancy on a smooth surface is a potential lc place.

In this note, we prove the semiampleness conjecture for Kawamata log terminal Calabi–Yau (CY) surface pairs over an excellent base ring. As applications, we deduce that generalized abundance and Serrano’s conjecture hold for surfaces. Finally, we study the semiampleness conjecture for CY threefolds over a mixed characteristic DVR.

We study lc pairs polarized by a nef and log big divisor. After proving the minimal model theory for projective lc pairs polarized by a nef and log big divisor, we prove the effectivity of the Iitaka fibrations and some boundedness results for dlt pairs polarized by a nef and log big divisor.

We establish adjunction and inversion of adjunction for log canonical centers of arbitrary codimension in full generality.

We prove rationality criteria over nonclosed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$. For the last type of such threefolds, we provide a unirationality criterion and construct examples of unirational but not stably rational varieties of this type.

Let $\Gamma $ be a finite set, and $X\ni x$ a fixed kawamata log terminal germ. For any lc germ $(X\ni x,B:=\sum _{i} b_iB_i)$, such that $b_i\in \Gamma $, Nakamura’s conjecture, which is equivalent to the ascending chain condition conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor E over $X\ni x$, such that $a(E,X,B)=\mathrm {mld}(X\ni x,B)$, and $a(E,X,0)$ is bounded from above. We extend Nakamura’s conjecture to the setting that $X\ni x$ is not necessarily fixed and $\Gamma $ satisfies the descending chain condition, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of $a(E,X,0)$ for any such E.

A conic bundle is a contraction $X\to Z$ between normal varieties of relative dimension $1$ such that $-K_X$ is relatively ample. We prove a conjecture of Shokurov that predicts that if $X\to Z$ is a conic bundle such that X has canonical singularities and Z is $\mathbb {Q}$ -Gorenstein, then Z is always $\frac {1}{2}$ -lc, and the multiplicities of the fibres over codimension $1$ points are bounded from above by $2$ . Both values $\frac {1}{2}$ and $2$ are sharp. This is achieved by solving a more general conjecture of Shokurov on singularities of bases of lc-trivial fibrations of relative dimension $1$ with canonical singularities.

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.