The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties.

In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$, a $G$-torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture.

Let $f_0$ and $f_1$ be two homogeneous polynomials of degree d in three complex variables $z_1,z_2,z_3$. We show that the Lê–Yomdin surface singularities defined by $g_0:=f_0+z_i^{d+m}$ and $g_1:=f_1+z_i^{d+m}$ have the same abstract topology, the same monodromy zeta-function, the same $\mu ^*$-invariant, but lie in distinct path-connected components of the $\mu ^*$-constant stratum if their projective tangent cones (defined by $f_0$ and $f_1$, respectively) make a Zariski pair of curves in $\mathbb {P}^2$, the singularities of which are Newton non-degenerate. In this case, we say that $V(g_0):=g_0^{-1}(0)$ and $V(g_1):=g_1^{-1}(0)$ make a $\mu ^*$-Zariski pair of surface singularities. Being such a pair is a necessary condition for the germs $V(g_0)$ and $V(g_1)$ to have distinct embedded topologies.

We prove that double Schubert polynomials have the saturated Newton polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study nonstandard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen–Macaulay prime ideal in a nonstandard multigrading, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid.

We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety.

Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties with nef tangent bundle and $T_M$ is numerically flat. We also prove that extremal contractions exist as smooth morphisms. As an application, we prove that, if the Frobenius morphism can be lifted modulo $p^2$, then $X$ admits, up to a finite étale Galois cover, a smooth morphism onto an ordinary abelian variety whose fibers are products of projective spaces.

We use the tropical geometry approach to compute absolute and relative enumerative invariants of complex surfaces which are $\mathbb {C} P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block–Göttsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to compute these invariants along with the associated interpretation as operators acting on some Fock space. The floor diagram algorithm allows one to prove the piecewise polynomiality of the relative invariants, and the quasi-modularity of their generating series.

Let $X_4\subset \mathbb {P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field k. We show that if either $X_4$ contains a linear subspace $\Lambda $ of dimension $h\geq \max \{2,\dim (\Lambda \cap \operatorname {\mathrm {Sing}}(X_4))+2\}$ or has double points along a linear subspace of dimension $h\geq 3$, a smooth k-rational point and is otherwise general, then $X_4$ is unirational over k. This improves previous results by A. Predonzan and J. Harris, B. Mazur and R. Pandharipande for quartics. We also provide a density result for the k-rational points of quartic $3$-folds with a double plane over a number field, and several unirationality results for quintic hypersurfaces over a $C_r$ field.

We show that a sufficiently general hypersurface of degree d in $\mathbb {P}^n$ admits a toric Gröbner degeneration after linear change of coordinates if and only if $d\leq 2n-1$.

We provide a criterion for a coherent sheaf to be an Ulrich sheaf in terms of a certain bilinear form on its global sections. When working over the real numbers, we call it a positive Ulrich sheaf if this bilinear form is symmetric or Hermitian and positive-definite. In that case, our result provides a common theoretical framework for several results in real algebraic geometry concerning the existence of algebraic certificates for certain geometric properties. For instance, it implies Hilbert’s theorem on nonnegative ternary quartics, via the geometry of del Pezzo surfaces, and the solution of the Lax conjecture on plane hyperbolic curves due to Helton and Vinnikov.

This papers classifies toric Fano threefolds with singular locus $\{ \frac {1}{k}(1,1,1) \}$ for $k \in \mathbb {Z}_{\geq 1}$ building on the work of Batyrev (1981, Nauk SSSR Ser. Mat. 45, 704–717) and Watanabe–Watanabe (1982, Tokyo J. Math. 5, 37–48). This is achieved by completing an equivalent problem in the language of Fano polytopes. Furthermore, we identify birational relationships between entries of the classification. For a fixed value $k \geq 4$, there are exactly two such toric Fano threefolds linked by a blowup in a torus-invariant line.

We generalize the works of Pappas–Rapoport–Zhu on twisted affine Grassmannians to the wildly ramified case under mild assumptions. This rests on a construction of certain smooth affine $\mathbb {Z}[t]$-groups with connected fibers of parahoric type, motivated by previous work of Tits. The resulting $\mathbb {F}_p(t)$-groups are pseudo-reductive and sometimes non-standard in the sense of Conrad–Gabber–Prasad, and their $\mathbb {F}_p [\hspace {-0,5mm}[ {t} ]\hspace {-0,5mm}] $-models are parahoric in a generalized sense. We study their affine Grassmannians, proving normality of Schubert varieties and Zhu’s coherence theorem.

A Richardson variety in a flag variety is an intersection of two Schubert varieties defined by transverse flags. We define and study relative Richardson varieties, which are defined over a base scheme with a vector bundle and two flags. To do so, we generalise transversality of flags to a relative notion, versality, that allows the flags to be non-transverse over some fibers. Relative Richardson varieties share many of the geometric properties of Richardson varieties. We generalise several geometric and cohomological facts about Richardson varieties to relative Richardson varieties. We also prove that the local geometry of a relative Richardson variety is governed, in a precise sense, by the two intersecting Schubert varieties, giving a generalisation, in the flag variety case, of a theorem of Knutson–Woo–Yong; we also generalise this result to intersections of arbitrarily many relative Schubert varieties. We give an application to Brill–Noether varieties on elliptic curves, and a conjectural generalisation to higher genus curves.

We introduce a conjecture on Virasoro constraints for the moduli space of stable sheaves on a smooth projective surface. These generalise the Virasoro constraints on the Hilbert scheme of a surface found by Moreira and Moreira, Oblomkov, Okounkov and Pandharipande. We verify the conjecture in many nontrivial cases by using a combinatorial description of equivariant sheaves found by Klyachko.

In this work, we prove a version of the Sylvester–Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of $\Sigma ^{[3]}\Pi \Sigma \Pi ^{[2]}$ circuits. Specifically, we prove that, if a finite set of irreducible quadratic polynomials ${\mathcal {Q}}$ satisfies that for every two polynomials $Q_1,Q_2\in {\mathcal {Q}}$ there is a subset ${\mathcal {K}}\subset {\mathcal {Q}}$ such that $Q_1,Q_2 \notin {\mathcal {K}}$ and whenever $Q_1$ and $Q_2$ vanish, then $\prod _{i\in {\mathcal {K}}} Q_i$ vanishes, then the linear span of the polynomials in ${\mathcal {Q}}$ has dimension $O(1)$. This extends the earlier result [21] that holds for the case $|{\mathcal {K}}| = 1$.

The most fundamental example of mirror symmetry compares the Fermat hypersurfaces in $\mathbb {P}^n$ and $\mathbb {P}^n/G$, where G is a finite group that acts on $\mathbb {P}^n$ and preserves the Fermat hypersurface. We generalize this to hypersurfaces in Grassmannians, where the picture is richer and more complex. There is a finite group G that acts on the Grassmannian $\operatorname {{\mathrm {Gr}}}(n,r)$ and preserves an appropriate Calabi–Yau hypersurface. We establish how mirror symmetry, toric degenerations, blow-ups and variation of GIT relate the Calabi–Yau hypersurfaces inside $\operatorname {{\mathrm {Gr}}}(n,r)$ and $\operatorname {{\mathrm {Gr}}}(n,r)/G$. This allows us to describe a compactification of the Eguchi–Hori–Xiong mirror to the Grassmannian, inside a blow-up of the quotient of the Grassmannian by G.

We give an upper bound on the volume $\operatorname {vol}(P^*)$ of a polytope $P^*$ dual to a d-dimensional lattice polytope P with exactly one interior lattice point in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp and achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree $(-K_X)^d$ of a d-dimensional Fano toric variety X with at worst canonical singularities.

It is well known that the diffeomorphism type of the Milnor fibration of a (Newton) nondegenerate polynomial function f is uniquely determined by the Newton boundary of f. In the present paper, we generalize this result to certain degenerate functions, namely we show that the diffeomorphism type of the Milnor fibration of a (possibly degenerate) polynomial function of the form $f=f^1\cdots f^{k_0}$ is uniquely determined by the Newton boundaries of $f^1,\ldots , f^{k_0}$ if $\{f^{k_1}=\cdots =f^{k_m}=0\}$ is a nondegenerate complete intersection variety for any $k_1,\ldots ,k_m\in \{1,\ldots , k_0\}$.

A famous problem in birational geometry is to determine when the birational automorphism group of a Fano variety is finite. The Noether–Fano method has been the main approach to this problem. The purpose of this paper is to give a new approach to the problem by showing that in every positive characteristic, there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphism group. To do this, we prove that these varieties admit ample and birationally equivariant line bundles. Our result applies the differential forms that Kollár produces on $p$-cyclic covers in characteristic $p > 0$.

We study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric Deligne-Mumford stacks (with possibly nontrivial generic stabilisers K and semi-projective coarse moduli spaces) relative to Lagrangian branes of Aganagic-Vafa type. An Aganagic-Vafa brane in this paper is a possibly ineffective $C^\infty $ orbifold that admits a presentation $[(S^1\times \mathbb {R} ^2)/G_\tau ]$, where $G_\tau $ is a finite abelian group containing K and $G_\tau /K \cong \boldsymbol {\mu }_{\mathfrak {m}}$ is cyclic of some order $\mathfrak {m}\in \mathbb {Z} _{>0}$.

1. 1. We present foundational materials of enumerative geometry of stable holomorphic maps from bordered orbifold Riemann surfaces to a 3-dimensional Calabi-Yau smooth toric DM stack $\mathcal {X}$ with boundaries mapped into an Aganagic-Vafa brane $\mathcal {L}$. All genus open-closed Gromov-Witten invariants of $\mathcal {X}$ relative to $\mathcal {L}$ are defined by torus localisation and depend on the choice of a framing $f\in \mathbb {Z} $ of $\mathcal {L}$.

2. 2. We provide another definition of all genus open-closed Gromov-Witten invariants in (1) based on algebraic relative orbifold Gromov-Witten theory, which agrees with the definition in (1) up to a sign depending on the choice of orientation on moduli of maps in (1). This generalises the definition in [57] for smooth toric Calabi-Yau 3-folds and specifies an orientation on moduli of maps in (1) compatible with the canonical orientation on moduli of relative stable maps determined by the complex structure.

3. 3. When $\mathcal {X}$ is a toric Calabi-Yau 3-orbifold (i.e., when the generic stabiliser K is trivial), so that $G_\tau =\boldsymbol {\mu }_{\mathfrak {m}}$, we define generating functions $F_{g,h}^{\mathcal {X},(\mathcal {L},f)}$ of open-closed Gromov-Witten invariants of arbitrary genus g and number h of boundary circles; it takes values in $H^*_{ {\mathrm {CR}} }(\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}; \mathbb {C} )^{\otimes h}$, where $H^*_{ {\mathrm {CR}} }(\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}; \mathbb {C} )\cong \mathbb {C} ^{\mathfrak {m}}$ is the Chen-Ruan orbifold cohomology of the classifying space $\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}$ of $\boldsymbol {\mu }_{\mathfrak {m}}$.

4. 4. We prove an open mirror theorem that relates the generating function $F_{0,1}^{\mathcal {X},(\mathcal {L},f)}$ of orbifold disk invariants to Abel-Jacobi maps of the mirror curve of $\mathcal {X}$. This generalises a conjecture by Aganagic-Vafa [6] and Aganagic-Klemm-Vafa [5] (proved in full generality by the first and the second authors in [33]) on the disk potential of a smooth semi-projective toric Calabi-Yau 3-fold.