The paper studies a relation between fundamental group of the complement to a plane singular curve and the orbifold pencils containing it. The main tool is the use of Albanese varieties of cyclic covers ramified along such curves. Our results give sufficient conditions for a plane singular curve to belong to an orbifold pencil, that is, a pencil of plane curves with multiple fibers inducing a map onto an orbifold curve whose orbifold fundamental group is nontrivial. We construct an example of a cyclic cover of the projective plane which is an abelian surface isomorphic to the Jacobian of a curve of genus 2 illustrating the extent to which these conditions are necessary.

We prove that the tautological ring of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$, the moduli space of $n$-pointed genus two curves of compact type, does not have Poincaré duality for any $n\geqslant 8$. This result is obtained via a more general study of the cohomology groups of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$. We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of $H^{k}({\mathcal{M}}_{2,n}^{\mathsf{ct}})$ for any $k$ and $n$ considered both as $\mathbb{S}_{n}$-representation and as mixed Hodge structure/$\ell$-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of $\overline{{\mathcal{M}}}_{2,n}$ is tautological for $n<20$, and that the tautological ring of $\overline{{\mathcal{M}}}_{2,n}$ fails to have Poincaré duality for all $n\geqslant 20$. This improves and simplifies results of the author and Orsola Tommasi.

In this paper we demonstrate that non-commutative localizations of arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category $\text{}\underline{C}$ by a set $S$ of morphisms in the heart $\text{}\underline{Hw}$ of a weight structure $w$ on it one obtains a triangulated category endowed with a weight structure $w^{\prime }$. The heart of $w^{\prime }$ is a certain version of the Karoubi envelope of the non-commutative localization $\text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ (of $\text{}\underline{Hw}$ by $S$). The functor $\text{}\underline{Hw}\rightarrow \text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ is the natural categorical version of Cohn’s localization of a ring, i.e., it is universal among additive functors that make all elements of $S$ invertible. For any additive category $\text{}\underline{A}$, taking $\text{}\underline{C}=K^{b}(\text{}\underline{A})$ we obtain a very efficient tool for computing $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$ coincides with the ‘abstract’ localization $\text{}\underline{A}[S^{-1}]$ (as constructed by Gabriel and Zisman) if $S$ contains all identity morphisms of $\text{}\underline{A}$ and is closed with respect to direct sums. We apply our results to certain categories of birational motives $DM_{gm}^{o}(U)$ (generalizing those defined by Kahn and Sujatha). We define $DM_{gm}^{o}(U)$ for an arbitrary $U$ as a certain localization of $K^{b}(Cor(U))$ and obtain a weight structure for it. When $U$ is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general $U$ the result is completely new. The existence of the corresponding adjacent$t$-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over $U$.

The Hilbert scheme $X^{[a]}$ of points on a complex manifold $X$ is a compactification of the configuration space of $a$ -element subsets of $X$ . The integral cohomology of $X^{[a]}$ is more subtle than the rational cohomology. In this paper, we compute the mod 2 cohomology of $X^{[2]}$ for any complex manifold $X$ , and the integral cohomology of $X^{[2]}$ when $X$ has torsion-free cohomology.

For each del Pezzo surface $S$ with du Val singularities, we determine whether it admits a $(-K_{S})$-polar cylinder or not. If it allows one, then we present an effective $\mathbb{Q}$-divisor $D$ that is $\mathbb{Q}$-linearly equivalent to $-K_{S}$ and such that the open set $S\setminus \text{Supp}(D)$ is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit non-trivial $\mathbb{G}_{a}$-actions on their affine cones defined by their anticanonical divisors.

We show that the subcategory of mixed Tate motives in Voevodsky’s derived category of motives is not closed under infinite products. In fact, the infinite product $\prod _{n=1}^{\infty }\mathbf{Q}(0)$ is not mixed Tate. More generally, the inclusions of several subcategories of motives do not have left or right adjoints. The proofs use the failure of finite generation for Chow groups in various contexts. In the positive direction, we show that for any scheme of finite type over a field whose motive is mixed Tate, the Chow groups are finitely generated.

The main aim of this paper is to classify Ulrich ideals and Ulrich modules over two-dimensional Gorenstein rational singularities (rational double points) from a geometric point of view. To achieve this purpose, we introduce the notion of (weakly) special Cohen–Macaulay modules with respect to ideals, and study the relationship between those modules and Ulrich modules with respect to good ideals.

We consider plane Cremona maps with proper base points and the base ideal generated by the linear system of forms defining the map. The object of this work is to study the link between the algebraic properties of the base ideal and those of the ideal of these points fattened by the virtual multiplicities arising from the linear system. We reveal conditions which naturally regulate this association, with particular emphasis on the homological side. While most classical numerical inequalities concern the three highest virtual multiplicities, here we emphasize also the role of one single highest multiplicity. In this vein we describe classes of Cremona maps for large and small values of the highest virtual multiplicity. We also deal with the delicate question as to when is the base ideal non-saturated and consider the structure of its saturation.

For a variety with a Whitney stratification by affine spaces, we study categories of motivic sheaves which are constant mixed Tate along the strata. We are particularly interested in those cases where the category of mixed Tate motives over a point is equivalent to the category of finite-dimensional bigraded vector spaces. Examples of such situations include rational motives on varieties over finite fields and modules over the spectrum representing the semisimplification of de Rham cohomology for varieties over the complex numbers. We show that our categories of stratified mixed Tate motives have a natural weight structure. Under an additional assumption of pointwise purity for objects of the heart, tilting gives an equivalence between stratified mixed Tate sheaves and the bounded homotopy category of the heart of the weight structure. Specializing to the case of flag varieties, we find natural geometric interpretations of graded category ${\mathcal{O}}$ and Koszul duality.

Using the inductive structure of a Fermat variety by Shioda and Katsura [‘On Fermat varieties’, Tohoku Math. J. (2) 31(1) (1979), 97–115], we estimate the refined motivic dimension of certain Fermat varieties. As an application of our computation, we present an elementary proof of the generalised Hodge conjecture for those varieties.

We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogs of Fourier–Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties.

We define a theory of étale motives over a noetherian scheme. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations of Grothendieck. The rational part of these categories coincides with the triangulated categories of Beilinson motives (and is thus strongly related to algebraic $K$-theory). We extend the rigidity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion étale motives essentially coincide with the usual complexes of torsion étale sheaves (at least if we restrict ourselves to torsion prime to the residue characteristics). As a consequence, we obtain the expected results of absolute purity, of finiteness, and of Grothendieck duality for étale motives with integral coefficients, by putting together their counterparts for Beilinson motives and for torsion étale sheaves. Following Thomason’s insights, this also provides a conceptual and convenient construction of the $\ell$-adic realization of motives, as the homotopy $\ell$-completion functor.

In this note we study properties of partially ample line bundles on simplicial projective toric varieties. We prove that the cone of q-ample line bundles is a union of rational polyhedral cones, and calculate these cones in examples. We prove a restriction theorem for big q-ample line bundles, and deduce that q-ampleness of the anticanonical bundle is not invariant under flips. Finally we prove a Kodaira-type vanishing theorem for q-ample line bundles.

Let τ be the involution changing the sign of two coordinates in ℙ4. We prove that τ induces the identity action on the second Chow group of the intersection of a τ-invariant cubic with a τ-invariant quadric hypersurface in ℙ4. Let lτ and Πτ be the one- and two-dimensional components of the fixed locus of the involution τ. We describe the generalized Prymian associated with the projection of a τ-invariant cubic 𝓵 ⊂ P4 from lτ onto Πτ in terms of the Prymians 𝓅2 and 𝓅3 associated with the double covers of two irreducible components, of degree 2 and 3, respectively, of the reducible discriminant curve. This gives a precise description of the induced action of the involution τ on the continuous part of the Chow group CH2 (𝓵). The action on the subgroup corresponding to 𝓅3 is the identity, and the action on the subgroup corresponding to 𝓅2 is the multiplication by —1.

For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

We give explicit formulas for the Hilbert series of residual intersections of a scheme in terms of the Hilbert series of its conormal modules. In a previous paper, we proved that such formulas should exist. We give applications to the number of equations defining projective varieties and to the dimension of secant varieties of surfaces and three-folds.

Consider a smooth quasi-projective variety $X$ equipped with a $\mathbb{C}^{\ast }$-action, and a regular function $f:X\rightarrow \mathbb{C}$ which is $\mathbb{C}^{\ast }$-equivariant with respect to a positive weight action on the base. We prove the purity of the mixed Hodge structure and the hard Lefschetz theorem on the cohomology of the vanishing cycle complex of $f$ on proper components of the critical locus of $f$, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities. Building on work by Kontsevich and Soibelman, Nagao, and Efimov, we use this result to prove the quantum positivity conjecture for cluster mutations for all quivers admitting a positively graded nondegenerate potential. We deduce quantum positivity for all quivers of rank at most 4; quivers with nondegenerate potential admitting a cut; and quivers with potential associated to triangulations of surfaces with marked points and nonempty boundary.

Assuming the Tate conjecture and the computability of étale cohomology with finite coefficients, we give an algorithm that computes the Néron–Severi group of any smooth projective geometrically integral variety, and also the rank of the group of numerical equivalence classes of codimension $p$ cycles for any $p$.

The goal of this paper is to prove that if certain ‘standard’ conjectures on motives over algebraically closed fields hold, then over any ‘reasonable’ scheme $S$ there exists a motivic$t$-structure for the category $\text{DM}_{c}(S)$ of relative Voevodsky’s motives (to be more precise, for the Beilinson motives described by Cisinski and Deglise). If $S$ is of finite type over a field, then the heart of this $t$-structure (the category of mixed motivic sheaves over $S$) is endowed with a weight filtration with semisimple factors. We also prove a certain ‘motivic decomposition theorem’ (assuming the conjectures mentioned) and characterize semisimple motivic sheaves over $S$ in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic $t$-structure is transversal to the Chow weight structure for $\text{DM}_{c}(S)$ (that was introduced previously by Hébert and the author). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow weight spectral sequences for ‘perverse étale homology’ (which we prove unconditionally); this statement also yields the existence of the Chow weight filtration for such (co)homology that is strictly restricted by (‘motivic’) morphisms.

Let $A$ be an abelian variety defined over a field $k$. In this paper we define a descending filtration $\{F^{r}\}_{r\geqslant 0}$ of the group $\mathit{CH}_{0}(A)$ and prove that the successive quotients $F^{r}/F^{r+1}\otimes \mathbb{Z}[1/r!]$ are isomorphic to the group $(K(k;A,\dots ,A)/Sym)\otimes \mathbb{Z}[1/r!]$, where $K(k;A,\dots ,A)$ is the Somekawa $K$-group attached to $r$-copies of the abelian variety $A$. In the special case when $k$ is a finite extension of $\mathbb{Q}_{p}$ and $A$ has split multiplicative reduction, we compute the kernel of the map $\mathit{CH}_{0}(A)\otimes \mathbb{Z}[\frac{1}{2}]\rightarrow \text{Hom}(Br(A),\mathbb{Q}/\mathbb{Z})\otimes \mathbb{Z}[\frac{1}{2}]$, induced by the pairing $\mathit{CH}_{0}(A)\times Br(A)\rightarrow \mathbb{Q}/\mathbb{Z}$.