We improve the Bend-and-Break result of Miyaoka and Mori by establishing the optimal degree bound. Our result also yields optimal bounds on lengths of extremal rays of log canonical pairs.

We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish. We obtain both negative and positive results, using unramified cohomology and birational rigidity techniques, as well as concrete rationality constructions.

We compute log canonical thresholds of reduced plane curves of degree $d$ at points of multiplicity $d-1$. As a consequence, we describe all possible values of log canonical threshold that are less than $2/(d-1)$ for reduced plane curves of degree $d$. In addition, we compute log canonical thresholds for all reduced plane curves of degree less than 6.

We discuss the relative log minimal model theory for log surfaces in the analytic setting. More precisely, we show that the minimal model program, the abundance theorem, and the finite generation of log canonical rings hold for log pairs of complex surfaces which are projective over complex analytic varieties.

The moduli space of bundle stable pairs $\overline {M}_C(2,\Lambda )$ on a smooth projective curve C, introduced by Thaddeus, is a smooth Fano variety of Picard rank two. Focusing on the genus two case, we show that its K-moduli space is isomorphic to a GIT moduli of lines in quartic del Pezzo threefolds. Additionally, we construct a natural forgetful morphism from the K-moduli of $\overline {M}_C(2,\Lambda )$ to that of the moduli spaces of stable vector bundles $\overline {N}_C(2,\Lambda )$. In particular, Thaddeus’ moduli spaces for genus two curves are all K-stable.

We show that a divisor in a rational homogenous variety with split normal sequence is the preimage of a hyperplane section in either the projective space or a quadric.

We show that a very general hypersurface of degree $d \geq 4$ and dimension $N \leq (d+1)2^{d-4}$ over a field of characteristic $\neq 2$ does not admit a decomposition of the diagonal; hence, it is neither stably nor retract rational, nor $\mathbb {A}^1$-connected. Similar results hold in characteristic $2$ under a slightly weaker degree bound. This improves earlier results in [44] and [33].

We correct an error in the statement of [LVW24, Theorem 3.2] by adding the omitted hypothesis that the covers are arithmetically Gorenstein.

For log canonical (lc) algebraically integrable foliations on Kawamata log terminal (klt) varieties, we prove the base-point-freeness theorem, the contraction theorem, and the existence of flips. The first result resolves a conjecture of Cascini and Spicer, while the latter two results strengthen a result of Cascini and Spicer by removing their assumption on the termination of flips. Moreover, we prove the existence of the minimal model program for lc algebraically integrable foliations on klt varieties and the existence of good minimal models or Mori fiber spaces for lc algebraically integrable foliations polarized by ample divisors on klt varieties. As a consequence, we show that $\mathbb{Q}$-factorial klt varieties with lc algebraically integrable Fano foliation structures are Mori dream spaces. We also show the existence of a Shokurov-type polytope for lc algebraically integrable foliations.

Let $\pi :X\rightarrow Z$ be a Fano type fibration with $\dim X-\dim Z=d$ and let $(X,B)$ be an $\epsilon $-lc pair with $K_X+B\sim _{\mathbb {R}} 0/Z$. The canonical bundle formula gives $(Z,B_Z+M_Z)$ where $B_Z$ is the discriminant divisor and $M_Z$ is the moduli divisor which is determined up to $\mathbb {R}$-linear equivalence. Shokurov conjectured that one can choose $M_Z\geqslant 0$ such that $(Z,B_Z+M_Z)$ is $\delta $-lc where $\delta $ only depends on $d,\epsilon $. Very recently, this conjecture was proved by Birkar [8]. For $d=1$ and $\epsilon =1$, Han, Jiang, and Luo [13] gave the optimal value of $\delta =1/2$. In this paper, we give the optimal value of $\delta $ for $d=1$ and arbitrary $0<\epsilon \leqslant 1$.

We construct pathological examples of MMP singularities in every positive characteristic using quotients by $\alpha _p$-actions. In particular, we obtain non-$S_3$ terminal singularities, as well as locally stable (respectively stable) families whose general fibers are smooth (respectively klt, Cohen–Macaulay, and F-injective) and whose special fibers are non-$S_2$. The dimensions of these examples are bounded below by a linear function of the characteristic.

While the splinter property is a local property for Noetherian schemes in characteristic zero, Bhatt observed that it imposes strong conditions on the global geometry of proper schemes in positive characteristic. We show that if a proper scheme over a field of positive characteristic is a splinter, then its Nori fundamental group scheme is trivial and its Kodaira dimension is negative. In another direction, Bhatt also showed that any splinter in positive characteristic is a derived splinter. We ask whether the splinter property is a derived invariant for projective varieties in positive characteristic and give a positive answer for normal Gorenstein projective varieties with big anticanonical divisor. We also show that global F-regularity is a derived invariant for normal Gorenstein projective varieties in positive characteristic.

We prove existence of flips for log canonical foliated pairs of rank one on a ${\mathbb Q}$-factorial projective klt threefold. This, in particular, provides a proof of the existence of a minimal model for a rank one foliation on a threefold for a wider range of singularities, after McQuillan.

We discuss a conjecture of Shokurov on the semi-ampleness of the moduli part of a general fibration.

It was conjectured by McKernan and Shokurov that for any Fano contraction $f:X \to Z$ of relative dimension r with X being $\epsilon $-lc, there is a positive $\delta $ depending only on $r,\epsilon $ such that Z is $\delta $-lc and the multiplicity of the fiber of f over a codimension one point of Z is bounded from above by $1/\delta $. Recently, this conjecture was confirmed by Birkar [9]. In this article, we give an explicit value for $\delta $ in terms of $\epsilon ,r$ in the toric case, which belongs to $O(\epsilon ^{2^r})$ as $\epsilon \rightarrow 0$. The order $O(\epsilon ^{2^r})$ is optimal in some sense.

We prove motivic versions of mass formulas by Krasner, Serre, and Bhargava concerning (weighted) counts of extensions of local fields.

Given a family of pairs over a smooth curve whose general fiber is a log Calabi–Yau pair in a fixed bounded family, suppose there exists a divisor on the family whose restriction on a general fiber is ample with bounded volume. We show that if the total space of the family has relatively trivial log canonical divisor and the special fiber has slc singularities, then every irreducible component of the special fiber is birationally bounded.

Let $(X,\Delta )$ be a normal pair with a projective morphism $X \to Z$ and let A be a relatively ample $\mathbb {R}$-divisor on X. We prove the termination of some minimal model program on $(X,\Delta +A)/Z$ and the abundance conjecture for its minimal model under assumptions that the non-nef locus of $K_{X}+\Delta +A$ over Z does not intersect the non-lc locus of $(X,\Delta )$ and that the restriction of $K_{X}+\Delta +A$ to the non-lc locus of $(X,\Delta )$ is semi-ample over Z.

We show that for any $n\geq5$ there exist connected algebraic subgroups in the Cremona group $\text{Bir}(\mathbb P^n)$ that are not contained in any maximal connected algebraic subgroup. Our approach exploits the existence of stably rational, non-rational threefolds.

We classify finite groups that act faithfully by symplectic birational transformations on an irreducible holomorphic symplectic (IHS) manifold of $OG10$ type. In particular, if X is an IHS manifold of $OG10$ type and G a finite subgroup of symplectic birational transformations of X, then the action of G on $H^2(X,\mathbb {Z})$ is conjugate to a subgroup of one of 375 groups of isometries. We prove a criterion for when such a group is determined by a group of automorphisms acting on a cubic fourfold, and apply it to our classification. Our proof is computer aided, and our results are available in a Zenodo dataset.