Let R be a commutative ring. One may ask when a general R-module P that satisfies $P \oplus R \cong R^n$ has a free summand of a given rank. M. Raynaud translated this question into one about sections of certain maps between Stiefel varieties: if $V_r(\mathbb {A}^n)$ denotes the variety $\operatorname {GL}(n) / \operatorname {GL}(n-r)$ over a field k, then the projection $V_r(\mathbb {A}^n) \to V_1(\mathbb {A}^n)$ has a section if and only if the following holds: any module P over any k-algebra R with the property that $P \oplus R \cong R^n$ has a free summand of rank $r-1$. Using techniques from $\mathbb {A}^1$-homotopy theory, we characterize those n for which the map $V_r(\mathbb {A}^n) \to V_1(\mathbb {A}^n)$ has a section in the cases $r=3,4$ under some assumptions on the base field.

We conclude that if $P \oplus R \cong R^{24m}$ and R contains the field of rational numbers, then P contains a free summand of rank $2$. If R contains a quadratically closed field of characteristic $0$, or the field of real numbers, then P contains a free summand of rank $3$. The analogous results hold for schemes and vector bundles over them.

A real variety whose real locus achieves the Smith–Thom equality is called maximal. This paper introduces new constructions of maximal real varieties, by using moduli spaces of geometric objects. We establish the maximality of the following real varieties:

• – moduli spaces of stable vector bundles of coprime rank and degree over a maximal real curve (recovering Brugallé–Schaffhauser’s theorem with a short new proof), which extends to moduli spaces of parabolic vector bundles;

• – moduli spaces of stable Higgs bundles of coprime rank and degree over a maximal real curve, providing maximal hyper-Kähler manifolds in every even dimension;

• – if a real variety has maximal Hilbert square, then the variety and its Hilbert cube are maximal, which happens for all maximal real cubic 3-folds, but never for maximal real cubic 4-folds;

• – punctual Hilbert schemes on a maximal real surface with vanishing first $\mathbb {F}_2$-Betti number and connected real locus, such as $\mathbb {R}$-rational maximal real surfaces and some generalized Dolgachev surfaces;

• – moduli spaces of stable sheaves on an $\mathbb {R}$-rational maximal Poisson surface (e.g. the real projective plane).

Let S be a minimal irregular surface of general type, whose Albanese map induces a hyperelliptic fibration $f:\,S \to B$ of genus g. We prove a quadratic upper bound on the genus g (i.e., $g\leq h\big (\chi (\mathcal {O}_S)\big )$, where h is a quadratic function). We also construct examples showing that the quadratic upper bounds cannot be improved to linear ones. In the special case when $p_g(S)=q(S)=1$, we show that $g\leq 14$.

Let $\mathcal {X}\to \mathbb {D}$ be a flat family of projective complex 3-folds over a disc $\mathbb {D}$ with smooth total space $\mathcal {X}$ and smooth general fibre $\mathcal {X}_t,$ and whose special fiber $\mathcal {X}_0$ has double normal crossing singularities, in particular, $\mathcal {X}_0=A\cup B$, with A, B smooth threefolds intersecting transversally along a smooth surface $R=A\cap B.$ In this paper, we first study the limit singularities of a $\delta $-nodal surface in the general fibre $S_t\subset \mathcal {X}_t$, when $S_t$ tends to the central fibre in such a way its $\delta $ nodes tend to distinct points in R. The result is that the limit surface $S_0$ is in general the union $S_0=S_A\cup S_B$, with $S_A\subset A$, $S_B\subset B$ smooth surfaces, intersecting on R along a $\delta $-nodal curve $C=S_A\cap R=S_B\cap B$. Then we prove that, under suitable conditions, a surface $S_0=S_A\cup S_B$ as above indeed deforms to a $\delta $-nodal surface in the general fibre of $\mathcal {X}\to \mathbb {D}$. As applications, we prove that there are regular irreducible components of the Severi variety of degree d surfaces with $\delta $ nodes in $\mathbb {P}^3$, for every $\delta \leqslant {d-1\choose 2}$ and of the Severi variety of complete intersection $\delta $-nodal surfaces of type $(d,h)$, with $d\geqslant h-1$ in $\mathbb {P}^4$, for every $\delta \leqslant {{d+3}\choose 3}-{{d-h+1}\choose 3}-1.$

We study the problem of the irreducibility of the Hessian variety ${\mathcal {H}}_f$ associated with a smooth cubic hypersurface $V(f)\subset {\mathbb {P}}^n$. We prove that when $n\leq 5$, ${\mathcal {H}}_f$ is normal and irreducible if and only if f is not of Thom-Sebastiani type (i.e., if one cannot separate its variables by changing coordinates). This also generalizes a result of Beniamino Segre dealing with the case of cubic surfaces. The geometric approach is based on the study of the singular locus of the Hessian variety and on infinitesimal computations arising from a particular description of these singularities.

Let k be an algebraically closed field of characteristic $p>0$. Let X be a normal projective surface over k with canonical singularities whose anticanonical divisor is nef and big. We prove that X is globally F-regular except for the following cases: (1) $K_X^2=4$ and $p=2$, (2) $K_X^2=3$ and $p \in \{2, 3\}$, (3) $K_X^2=2$ and $p \in \{2, 3\}$, (4) $K_X^2=1$ and $p \in \{2, 3, 5\}$. For each degree $K_X^2$, the assumption of p is optimal.

We prove a genus zero Givental-style mirror theorem for all complete intersections in toric Deligne-Mumford stacks, which provides an explicit slice called big I-function on Givental’s Lagrangian cone for such targets. In particular, we remove a technical assumption called convexity needed in the previous mirror theorem for such complete intersections. In the realm of quasimap theory, our mirror theorem can be viewed as solving the quasimap wall-crossing conjecture for big I-function [13] for these targets. In the proof, we discover a new recursive characterization of the slice on Givental’s Lagrangian cone, which may be of self-independent interests.

In this paper, we prove that if a three-dimensional quasi-projective variety X over an algebraically closed field of characteristic $p>3$ has only log canonical singularities, then so does a general hyperplane section H of X. We also show that the same is true for klt singularities, which is a slight extension of [15]. In the course of the proof, we provide a sufficient condition for log canonical (resp. klt) surface singularities to be geometrically log canonical (resp. geometrically klt) over a field.

For a smooth projective surface $X$ satisfying $H_1(X,\mathbb{Z}) = 0$ and $w \in H^2(X,\mu _r)$, we study deformation invariants of the pair $(X,w)$. Choosing a Brauer–Severi variety $Y$ (or, equivalently, Azumaya algebra $\mathcal{A}$) over $X$ with Stiefel–Whitney class $w$, the invariants are defined as virtual intersection numbers on suitable moduli spaces of stable twisted sheaves on $Y$ constructed by Yoshioka (or, equivalently, moduli spaces of $\mathcal{A}$-modules of Hoffmann–Stuhler).

We show that the invariants do not depend on the choice of $Y$. Using a result of de Jong, we observe that they are deformation invariants of the pair $(X,w)$. For surfaces with $h^{2,0}(X) \gt 0$, we show that the invariants can often be expressed as virtual intersection numbers on Gieseker–Maruyama–Simpson moduli spaces of stable sheaves on $X$. This can be seen as a ${\rm PGL}_r$–${\rm SL}_r$ correspondence.

As an application, we express ${\rm SU}(r) / \mu _r$ Vafa–Witten invariants of $X$ in terms of ${\rm SU}(r)$ Vafa–Witten invariants of $X$. We also show how formulae from Donaldson theory can be used to obtain upper bounds for the minimal second Chern class of Azumaya algebras on $X$ with given division algebra at the generic point.

We determine the cones of effective and nef divisors on the toroidal compactification of the ball quotient model of the moduli space of complex cubic surfaces with a chosen line. From this we also compute the corresponding cones for the moduli space of unmarked cubic surfaces.

Given a general polarized $K3$ surface $S\subset \mathbb P^g$ of genus $g\le 14$, we study projections of minimal degree and their variational structure. In particular, we prove that the degree of irrationality of all such surfaces is at most $4$, and that for $g=7,8,9,11$ there are no rational maps of degree $3$ induced by the primitive linear system. Our methods combine vector bundle techniques à la Lazarsfeld with derived category tools and also make use of the rich theory of singular curves on $K3$ surfaces.

Mukai’s program in [16] seeks to recover a K3 surface X from any curve C on it by exhibiting it as a Fourier–Mukai partner to a Brill–Noether locus of vector bundles on the curve. In the case X has Picard number one and the curve $C\in |H|$ is primitive, this was confirmed by Feyzbakhsh in [11, 13] for $g\geq 11$ and $g\neq 12$. More recently, Feyzbakhsh has shown in [12] that certain moduli spaces of stable bundles on X are isomorphic to the Brill–Noether locus of curves in $|H|$ if g is sufficiently large. In this paper, we work with irreducible curves in a nonprimitive ample linear system $|mH|$ and prove that Mukai’s program is valid for any irreducible curve when $g\neq 2$, $mg\geq 11$ and $mg\neq 12$. Furthermore, we introduce the destabilising regions to improve Feyzbakhsh’s analysis in [12]. We show that there are hyper-Kähler varieties as Brill–Noether loci of curves in every dimension.

We describe a geometric, stable pair compactification of the moduli space of Enriques surfaces with a numerical polarization of degree $2$, and identify it with a semitoroidal compactification of the period space.

In this article, we discuss the topology of varieties over $\mathbb {C}$, viz., their homology and homotopy groups. We show that the fundamental group of a quasi-projective variety has negative deficiency under a certain hypothesis on its second homology and therefore a large class of groups cannot arise as fundamental groups of varieties. For a smooth projective surface admitting a fibration over a curve, we give a detailed analysis of the homology and homotopy groups of their universal cover via a case-by-case analysis, depending on the nature of the singular fibers. For smooth, projective surfaces whose universal cover is holomorphically convex (conjecturally always true), we show that the second and third homotopy groups are free abelian, often of infinite rank.

In this article, we give a definition of weak stability condition on a triangulated category. The difference between our definition and existing definitions is that we allow objects in the kernel to have non-maximal phases. We then construct four types of weak stability conditions that naturally occur on Weierstraß ellitpic surfaces as limites of Bridgeland stability conditions.

In 2002, Fukaya [19] proposed a remarkable explanation of mirror symmetry detailing the Strominger–Yau–Zaslow (SYZ) conjecture [47] by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi–Yau manifold $\check {X}$ and the multivalued Morse theory on the base $\check {B}$ of an SYZ fibration $\check {p}\colon \check {X}\to \check {B}$, and the other between deformation theory of the mirror X and the same multivalued Morse theory on $\check {B}$. In this paper, we prove a reformulation of the main conjecture in Fukaya’s second correspondence, where multivalued Morse theory on the base $\check {B}$ is replaced by tropical geometry on the Legendre dual B. In the proof, we apply techniques of asymptotic analysis developed in [7, 9] to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi–Yau log variety introduced in [8]. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semiflat part $X_{\mathrm {sf}} \subset X$ allows us to extract consistent scattering diagrams from appropriate Maurer–Cartan solutions.

We use deformations and mutations of scattering diagrams to show that a scattering diagram with initial functions $f_1=(1+tx)^\mu $ and $f_2=(1+ty)^\nu $ has a dense region. This answers a question asked by Gross and Pandharipande [‘Quivers, curves, and the tropical vertex’, Port. Math. 67(2) (2010), 211–259] which had been proved only for the case $\mu =\nu $.

Building upon the classification by Lacini, we determine the isomorphism classes of log del Pezzo surfaces of rank one over an algebraically closed field of characteristic five either which are not log liftable over the ring of Witt vectors or whose singularities are not feasible in characteristic zero. We also show that such a surface is always constructed from the Du Val del Pezzo surface of Dynkin type $2[2^4]$. Furthermore, We show that the Kawamata–Viehweg vanishing theorem for ample $\mathbb {Z}$-Weil divisors holds for log del Pezzo surfaces of rank one in characteristic five if those singularities are feasible in characteristic zero.

The delta invariant interprets the criterion for the K-(poly)stability of log terminal Fano varieties. In this paper, we determine local delta invariants for all weak del Pezzo surfaces with the anti-canonical degree $\geq 5$.

Let X be a smooth projective variety of dimension $n\geq 2$ and $G\cong \mathbf {Z}^{n-1}$ a free abelian group of automorphisms of X over $\overline {\mathbf {Q}}$. Suppose that G is of positive entropy. We construct a canonical height function $\widehat {h}_G$ associated with G, corresponding to a nef and big $\mathbf {R}$-divisor, satisfying the Northcott property. By characterizing the zero locus of $\widehat {h}_G$, we prove the Kawaguchi–Silverman conjecture for each element of G. As for other applications, we determine the height counting function for non-periodic points and show that X satisfies potential density.