We determine the algebraic and transcendental lattices of a general cubic fourfold with a symplectic automorphism of prime order. We prove that cubic fourfolds admitting a symplectic automorphism of order at least three are rational, and we exhibit two families of rational cubic fourfolds that are not equivariantly rational with respect to their group of automorphisms. As an application, we determine the cohomological action of symplectic birational transformations of manifolds of OG10 type that are induced by prime order symplectic automorphisms of cubic fourfolds.

Under the assumption that the adjusted Brill-Noether number $\widetilde {\rho }$ is at least $-g$, we prove that the Brill-Noether loci in ${\mathcal M}_{g,n}$ of pointed curves carrying pencils with prescribed ramification at the marked points have a component of the expected codimension with pointed curves having Brill-Noether varieties of pencils of the minimal dimension. As an application, the map from the Hurwitz scheme to ${\mathcal M}_g$ is dominant if $n+\widetilde {\rho } \geq 0$ and generically finite otherwise, settling a variation of a classical problem of Zariski.

In the second part of the paper, we study the analogous loci of curves in Severi varieties on $K3$ surfaces, proving existence of curves with nongeneral behaviour from the point of view of Brill-Noether theory. This extends previous results of Ciliberto and the first-named author to the ramified case. We apply these results to study correspondences and cycles on $K3$ surfaces in relation to Beauville-Voisin points and constant cycle curves.

In this article, we classify irregular threefolds with numerically trivial canonical divisors in positive characteristic. For a threefold, if its Albanese dimension is not maximal, then the Albanese morphism will induce a fibration which either maps to a curve or is fibered by curves. In practice, we treat arbitrary dimensional irregular varieties with either one-dimensional Albanese fiber or one-dimensional Albanese image. We prove that such a variety carries another fibration transversal to its Albanese morphism (a “bi-fibration” structure), which is an analog structure of bielliptic or quasi-bielliptic surfaces. In turn, we give an explicit description of irregular threefolds with trivial canonical divisors.

We present an explicit geometric invariant theory construction which produces both the minimal resolution of the type $D_4$ surface singularity, and also the orbifold resolution. Our construction is based on a Tannakian approach which is in principle applicable to larger groups.

The compactification $\overline {\mathfrak M}_{1,3}$ of the Gieseker moduli space of surfaces of general type with $K_X^2 =1 $ and $\chi (X)=3$ in the moduli space of stable surfaces parametrises the so-called stable I-surfaces.

We classify all such surfaces which are 2-Gorenstein into four types using a mix of algebraic and geometric techniques. We find a new divisor in the closure of the Gieseker component and a new irreducible component of the moduli space.

We consider the countably many families $\mathcal {L}_d$, $d\in \mathbb {N}_{\geq 2}$, of K3 surfaces admitting an elliptic fibration with positive Mordell–Weil rank. We prove that the elliptic fibrations on the very general member of these families have the potential Mordell–Weil rank jump property for $d\neq 2,3$ and moreover the Mordell–Weil rank jump property for $d\equiv 3\ \mod 4$, $d\neq 3$. We provide explicit examples and discuss some extensions to subfamilies. The result is based on the geometric interaction between the (potential) Mordell–Weil rank jump property and the presence of special multisections of the fibration.

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 show that each connected component of the moduli space of smooth real binary quintics is isomorphic to an open subset of an arithmetic quotient of the real hyperbolic plane. Moreover, our main result says that the induced metric on this moduli space extends to a complete real hyperbolic orbifold structure on the space of stable real binary quintics. This turns the moduli space of stable real binary quintics into the quotient of the real hyperbolic plane by an explicit non-arithmetic triangle group.

In this paper, we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points $\{e_i-e_j \ \vert \ i \neq j\} \cup \{\pm e_i\}$ in $\mathbb {R}^n$, where $e_1,\dots ,e_n$ is the standard basis of $\mathbb {R}^n$. Such a polytope can be encoded by a quiver Q with vertices $V \subseteq \{{\upsilon }_1,\dots ,{\upsilon }_n\} \cup \{\star \}$, where each edge ${\upsilon }_j\to {\upsilon }_i$ or $\star \to {\upsilon }_i$ or ${\upsilon }_i\to \star $ gives rise to the point $e_i-e_j$ or $e_i$ or $-e_i$, respectively; we denote the corresponding polytope as $\operatorname {Root}(Q)$. These polytopes have been studied extensively under names such as edge polytope and root polytope. We show that if the quiver Q is strongly-connected, then the root polytope $\operatorname {Root}(Q)$ is reflexive and terminal; we moreover give a combinatorial description of the facets of $\operatorname {Root}(Q)$. We also show that if Q is planar, then $\operatorname {Root}(Q)$ is (integrally equivalent to) the polar dual of the flow polytope of the planar dual quiver $Q^{\vee }$. Finally, we consider the case that Q comes from the Hasse diagram of a finite ranked poset P and show in this case that $\operatorname {Root}(Q)$ is polar dual to (a translation of) a marked order polytope. We then go on to study the toric variety $Y(\mathcal {F}_Q)$ associated to the face fan $\mathcal {F}_Q$ of $\operatorname {Root}(Q)$. If Q comes from a ranked poset P, we give a combinatorial description of the Picard group of $Y(\mathcal {F}_Q)$, in terms of a new canonical ranked extension of P, and we show that $Y(\mathcal {F}_Q)$ is a small partial desingularisation of the Hibi projective toric variety $Y_{\mathcal {O}(P)}$ of the order polytope $\mathcal {O}(P)$. We show that $Y(\mathcal {F}_Q)$ has a small crepant toric resolution of singularities $Y(\widehat {\mathcal {F}}_Q)$ and, as a consequence that the Hibi toric variety $Y_{\mathcal {O}(P)}$ has a small resolution of singularities for any ranked poset P. These results have applications to mirror symmetry [61].

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.

Segre and Verlinde series have been studied in many cases, including virtual geometries of Quot schemes on surfaces and Calabi–Yau 4-folds. Our work is the first to address the equivariant setting for both ${\mathbb{C}}^2$ and ${\mathbb{C}}^4$ by examining higher degree contributions which have no compact analogue.

1. (i) For ${\mathbb{C}}^2$, we work mostly with virtual geometries of Quot schemes. After connecting the equivariant series in degree zero to the existing results of the first author for compact surfaces, we extend the Segre–Verlinde correspondence to all degrees and to the reduced virtual classes. Additionally, we conjecture that there is an equivariant symmetry of Segre series, which was also observed in the compact setting.

2. (ii) For ${\mathbb{C}}^4$, we give further motivation for the definition of the Verlinde series. Based on empirical data andtorsiopn additional structural results, we conjecture that there is an equivariant Segre–Verlinde correspondence and Segre symmetry analogous to the one for ${\mathbb{C}}^2$.

We prove that for every relatively prime pair of integers $(d,r)$ with $r>0$, there exists an exceptional pair $({\mathcal {O}},V)$ on any del Pezzo surface of degree $4$, such that V is a bundle of rank r and degree d. As an application, we prove that every Feigin-Odesskii Poisson bracket on a projective space can be included into a $5$-dimensional linear space of compatible Poisson brackets. We also construct new examples of linear spaces of compatible Feigin-Odesskii Poisson brackets of dimension $>5$, coming from del Pezzo surfaces of degree $>4$.

We prove that every irreducible component of the coarse Kollár-Shepherd-Barron and Alexeev (KSBA) moduli space of stable log Calabi–Yau surfaces admits a finite cover by a projective toric variety. This verifies a conjecture of Hacking–Keel–Yu. The proof combines tools from log smooth deformation theory, the minimal model program, punctured log Gromov–Witten theory, and mirror symmetry.

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.

For a klt singularity, C. Xu and Z. Zhuang [33] proved the associated graded algebra of a minimizing valuation of the normalized volume function is finitely generated, finishing the proof of the stable degeneration conjecture proposed by C. Li and C. Xu. We prove a family version of the stable degeneration: for a locally stable family of klt singularities with constant local volume, the ideal sequences of the minimizing valuations for the normalized volume function form families of ideals with flat cosupport, which induce a degeneration to a locally stable family of K-semistable log Fano cone singularities. In the proof, we give a method to construct families of Kollár models, which are a crucial tool introduced by Xu–Zhuang to prove finite generation for valuations of higher rational rank.

We study the moduli space of constant scalar curvature Kähler (cscK) surfaces around toric surfaces. To this end, we introduce the class of foldable surfaces: smooth toric surfaces whose lattice automorphism group contains a non-trivial cyclic subgroup. We classify such surfaces and show that they all admit a cscK metric. We then study the moduli space of polarised cscK surfaces around a point given by a foldable surface, and show that it is locally modelled on a finite quotient of a toric affine variety with terminal singularities.

In this article, we apply the Bestvina–Mess type formula for relatively hyperbolic groups, which is established by Tomohiro Fukaya, to automorphism groups of K3 surfaces, and we show that the virtual cohomological dimension of automorphism groups of K3 surfaces is determined by the covering dimension of the blown-up boundaries associated with their ample cones.

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 determine explicit generators for the ring of modular forms associated with the moduli spaces of K3 surfaces with automorphism group $(\mathbb {Z}/2\mathbb {Z})^2$ and of Picard rank 13 and higher. The K3 surfaces in question carry a canonical Jacobian elliptic fibration and the modular form generators appear as coefficients in the Weierstrass-type equations describing these fibrations.