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.

We conjecture that the exceptional set in Manin's conjecture has an explicit geometric description. Our proposal includes the rational point contributions from any generically finite map with larger geometric invariants. We prove that this set is contained in a thin subset of rational points, verifying that there is no counterexample to Manin's conjecture which arises from an incompatibility of geometric invariants.

The aim of this paper is to study all the natural first steps of the minimal model program for the moduli space of stable pointed curves. We prove that they admit a modular interpretation, and we study their geometric properties. As a particular case, we recover the first few Hassett–Keel log canonical models. As a by-product, we produce many birational morphisms from the moduli space of stable pointed curves to alternative modular projective compactifications of the moduli space of pointed curves.

We establish two results on three-dimensional del Pezzo fibrations in positive characteristic. First, we give an explicit bound for torsion index of relatively torsion line bundles. Second, we show the existence of purely inseparable sections with explicit bounded degree. To prove these results, we study log del Pezzo surfaces defined over imperfect fields.

We develop some foundational results in a higher-dimensional foliated Mori theory, and show how these results can be used to prove a structure theorem for the Kleiman–Mori cone of curves in terms of the numerical properties of $K_{{\mathcal{F}}}$ for rank 2 foliations on threefolds. We also make progress toward realizing a minimal model program (MMP) for rank 2 foliations on threefolds.

We construct non-archimedean SYZ (Strominger–Yau–Zaslow) fibrations for maximally degenerate Calabi–Yau varieties, and we show that they are affinoid torus fibrations away from a codimension-two subset of the base. This confirms a prediction by Kontsevich and Soibelman. We also give an explicit description of the induced integral affine structure on the base of the SYZ fibration. Our main technical tool is a study of the structure of minimal dlt (divisorially log terminal) models along one-dimensional strata.

We complete Mori's program with symmetric divisors for the moduli space of stable seven-pointed rational curves. We describe all birational models in terms of explicit blow-ups and blow-downs. We also give a moduli theoretic description of the first flip, which has not appeared in the literature.

In this paper we prove that a smooth family of canonically polarized manifolds parametrized by a special (in the sense of Campana) quasi-projective variety is isotrivial.

Assuming a particular case of the Borisov–Alexeev–Borisov conjecture, we prove that finite subgroups of the automorphism group of a finitely generated field over $\mathbb{Q}$ have bounded orders. Further, we investigate which algebraic varieties have groups of birational selfmaps satisfying the Jordan property. Unless explicitly stated otherwise, all varieties are assumed to be algebraic, geometrically irreducible and defined over an arbitrary field $\Bbbk$ of characteristic zero.

We show that the minimal model program on any smooth projective surface is realized as a variation of the moduli spaces of Bridgeland stable objects in the derived category of coherent sheaves.

We prove the finiteness of log pluricanonical representations for projective log canonical pairs with semi-ample log canonical divisor. As a corollary, we obtain that the log canonical divisor of a projective semi log canonical pair is semi-ample if and only if the log canonical divisor of its normalization is semi-ample. We also treat many other applications.

We prove a result on the inversion of adjunction for log canonical pairs that generalizes Kawakita's result to log canonical centres of arbitrary codimension.

The Sarkisov program studies birational maps between varieties that are end products of the Minimal Model Program (MMP) on nonsingular uniruled varieties. If $X$ and $Y$ are terminal $ \mathbb{Q} $-factorial projective varieties endowed with a structure of Mori fibre space, a birational map $f: X\dashrightarrow Y$ is the composition of a finite number of elementary Sarkisov links. This decomposition is in general not unique: two such define a relation in the Sarkisov program. I define elementary relations, and show they generate relations in the Sarkisov program. Roughly speaking, elementary relations are the relations among the end products of suitable relative MMPs of $Z$ over $W$ with $\rho (Z/ W)= 3$.

Let $(X,D)$ be a dlt pair, where $X$ is a normal projective variety. We show that any smooth family of canonically polarized varieties over $X\setminus \,{\rm Supp}\lfloor D \rfloor $ is isotrivial if the divisor $-(K_X+D)$ is ample. This result extends results of Viehweg–Zuo and Kebekus–Kovács. To prove this result we show that any extremal ray of the moving cone is generated by a family of curves, and these curves are contracted after a certain run of the minimal model program. In the log Fano case, this generalizes a theorem by Araujo from the klt to the dlt case. In order to run the minimal model program, we have to switch to a $\mathbb Q$-factorialization of $X$. As $\mathbb Q$-factorializations are generally not unique, we use flops to pass from one $\mathbb Q$-factorialization to another, proving the existence of a $\mathbb Q$-factorialization suitable for our purposes.

We prove that the moduli spaces of n-pointed m-stable curves introduced in our previous paper have projective coarse moduli. We use the resulting spaces to run an analogue of Hassett’s log minimal model program for .

Let $X$ be a minuscule Schubert variety. In this paper, we associate a quiver with $X$ and use the combinatorics of this quiver to describe all relative minimal models $\widehat{\pi}:{\widehat{X}}\to X$. We prove that all the morphisms $\widehat{\pi}$ are small and give a combinatorial criterion for $\widehat{X}$ to be smooth and thus a small resolution of $X$. We describe in this way all small resolutions of $X$. As another application of this description of relative minimal models, we obtain the following more intrinsic statement of the main result of Perrin, J. Algebra 294 (2005), 431–462. Let $\alpha\in A_1(X)$ be an effective 1-cycle class. Then the irreducible components of the scheme Hom$_{\alpha}(p^1,X)$ of morphisms from $\mathbb{P}^1$ to $X$ and of class $\alpha$ are indexed by the set: ${\mathfrak{ne}}(\alpha)=\{\beta\in A_1(\widehat{X}) \mid \beta$ is effective and $\widehat{\pi}_*\beta=\alpha\}$ which is independent of the choice of a relative minimal model $\widehat{X}$ of $X$.

Using the $\sharp$-minimal model program of uniruled varieties we show that, for any pair $(X, {\mathcal H})$ consisting of a reduced and irreducible variety $X$ of dimension $k \geq 3$ and a globally generated big line bundle ${\mathcal H}$ on $X$ with $d:= {\mathcal H}^k$ and $n:= h^0(X, {\mathcal H})-1$ such that $d<2(n-k)-4$, then $X$ is uniruled of ${\mathcal H}$-degree one, except if $(k,d,n)=(3,27,19)$ and a ${\sharp}$-minimal model of $(X, {\mathcal H})$ is $({\mathbb P}^3,{\mathcal O}_{{\mathbb P}^3}(3))$. We also show that the bound is optimal for threefolds.