We describe an equivalence between the notion of balanced twisted curve introduced by Abramovich and Vistoli, and a new notion of log twisted curve, which is a nodal curve equipped with some logarithmic data in the sense of Fontaine and Illusie. As applications of this equivalence, we construct a universal balanced twisted curve, prove that a balanced twisted curve over a general base scheme admits étale locally on the base a finite flat cover by a scheme, and also give a new construction of the moduli space of stable maps into a Deligne–Mumford stack and a new proof that it is bounded.

We obtain, via the formalism of tensor actions, a complete classification of the localizing subcategories of the stable derived category of any affine scheme that has hypersurface singularities or is a complete intersection in a regular scheme; in particular, this classifies the thick subcategories of the singularity categories of such rings. The analogous result is also proved for certain locally complete intersection schemes. It is also shown that from each of these classifications one can deduce the (relative) telescope conjecture.

We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao’s solution to Varchenko’s conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate the variety of critical points to the Chern–Schwartz–MacPherson class. The strengthened version recovers the geometric deletion–restriction formula of Denham et al. for arrangement complements, and generalizes Kouchnirenko’s theorem on the Newton polytope for nondegenerate hypersurfaces.

An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

We show that the algebraic local fundamental group of any Kawamata log terminal singularity as well as the algebraic fundamental group of the smooth locus of any log Fano variety are finite.

In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erdős and Turan. More precisely, given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. The method we use is deterministic: in particular, we do not assume independence or equidistribution of the coefficients of the polynomial.

We study quivers with potential (QPs) whose Jacobian algebras are finite-dimensional selfinjective. They are an analogue of the ‘good QPs’ studied by Bocklandt whose Jacobian algebras are 3-Calabi–Yau. We show that 2-representation-finite algebras are truncated Jacobian algebras of selfinjective QPs, which are factor algebras of Jacobian algebras by certain sets of arrows called cuts. We show that selfinjectivity of QPs is preserved under iterated mutation with respect to orbits of the Nakayama permutation. We give a sufficient condition for all truncated Jacobian algebras of a fixed QP to be derived equivalent. We introduce planar QPs which provide us with a rich source of selfinjective QPs.

We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We strengthen some theorems of F. Knop and S. Sahi and give two explicit formulas for these polynomials: a q-integral representation and a combinatorial formula. Our main tool is a q-integral representation for ordinary Macdonald polynomial. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials.

We study bounds for the Castelnuovo–Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular, our aim is to give a positive answer to a question posed by Bayer and Mumford in What can be computed in algebraic geometry? (Computational algebraic geometry and commutative algebra, Symposia Mathematica, vol. XXXIV (1993), 1–48) by showing that the known upper bound in characteristic zero holds true also in positive characteristic. We first analyse Giusti's proof, which provides the result in characteristic zero, giving some insight into the combinatorial properties needed in that context. For the general case, we provide a new argument which employs the Bayer–Stillman criterion for detecting regularity.

We give elementary proofs of the main theorems about the (small) quantum cohomology of Grassmannians, including the quantum Giambelli and quantum Pieri formulas, the rim-hook algorithm, the presentation, and a recent theorem of Fulton and Woodward about the minimal q-power which appears in a product of two Schubert classes.

In this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongly exceptional sequences of invertible sheaves on rational surfaces. We construct full strongly exceptional sequences for a large class of rational surfaces. For the case of toric surfaces we give a complete classification of full strongly exceptional sequences of invertible sheaves.

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPick(A) is the group of triangle auto-equivalences of Db(mod A) induced by two-sided tilting complexes. We study the group DPick(A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPick, as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPick(A) on a certain infinite quiver Γirr. This representation is faithful when the quiver Δ of A is a tree, and then DPick(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPick(A). When A is hereditary, DPick(A) coincides with the full group of k-linear triangle auto-equivalences of Db(mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to Db(mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.

As the simplest case of Langlands functoriality, one expects the existence of the symmetric power $S^n(\pi )$, where $\pi $ is an automorphic representation of ${\rm GL}(2,{\mathbb{A}})$ and ${\mathbb{A}}$ denotes the adeles of a number field $F$. This should be an automorphic representation of ${\rm GL}(N,{\mathbb{A}})$ ($N=n+1)$. This is known for $n=2,3$ and $4$. In this paper we show how to deduce the general case from a recent result of J.T. on deformation theory for ‘Schur representations’, combined with expected results on level-raising, as well as another case (a particular tensor product) of Langlands functoriality. Our methods assume $F$ totally real, and the initial representation $\pi $ of classical type.

We prove the μ-part of the main conjecture for modular forms along the anticyclotomic Zp-extension of a quadratic imaginary field. Our proof consists of first giving an explicit formula for the algebraic μ-invariant, and then using results of Ribet and Takahashi showing that our formula agrees with Vatsal’s formula for the analytic μ-invariant.

In this work, we study the deformation theory of ${\mathcal {E}}_n$-rings and the ${\mathcal {E}}_n$ analogue of the tangent complex, or topological André–Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence $A[n-1] \rightarrow T_A\rightarrow {\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)[n]$, relating the ${\mathcal {E}}_n$-tangent complex and ${\mathcal {E}}_n$-Hochschild cohomology of an ${\mathcal {E}}_n$-ring $A$. We give two proofs: the first is direct, reducing the problem to certain stable splittings of configuration spaces of punctured Euclidean spaces; the second is more conceptual, where we identify the sequence as the Lie algebras of a fiber sequence of derived algebraic groups, $B^{n-1}A^\times \rightarrow {\mathrm {Aut}}_A\rightarrow {\mathrm {Aut}}_{{\mathfrak B}^n\!A}$. Here ${\mathfrak B}^n\!A$ is an enriched $(\infty ,n)$-category constructed from $A$, and ${\mathcal {E}}_n$-Hochschild cohomology is realized as the infinitesimal automorphisms of ${\mathfrak B}^n\!A$. These groups are associated to moduli problems in ${\mathcal {E}}_{n+1}$-geometry, a less commutative form of derived algebraic geometry, in the sense of the work of Toën and Vezzosi and the work of Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital ${\mathcal {E}}_{n+1}$-algebra structure; in particular, the shifted tangent complex $T_A[-n]$ is a nonunital ${\mathcal {E}}_{n+1}$-algebra. The ${\mathcal {E}}_{n+1}$-algebra structure of this sequence extends the previously known ${\mathcal {E}}_{n+1}$-algebra structure on ${\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)$, given in the higher Deligne conjecture. In order to establish this moduli-theoretic interpretation, we make extensive use of factorization homology, a homology theory for framed $n$-manifolds with coefficients given by ${\mathcal {E}}_n$-algebras, constructed as a topological analogue of Beilinson and Drinfeld’s chiral homology. We give a separate exposition of this theory, developing the necessary results used in our proofs.

In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being ℚ-Gorenstein or the pair being log ℚ-Gorenstein. The main features of the theory extend to this setting in a natural way.

We construct a general semiregularity map for algebraic cycles as asked for by S. Bloch in 1972. The existence of such a semiregularity map has well known consequences for the structure of the Hilbert scheme and for the variational Hodge conjecture. Aside from generalizing and extending considerably previously known results in this direction, we give new applications to deformations of modules that encompass, for example, results of Artamkin and Mukai. The formation of the semiregularity map here involves powers of the cotangent complex, Atiyah classes, and trace maps, and is defined not only for subspaces of manifolds but for perfect complexes on arbitrary complex spaces. It generalizes in particular Illusie's treatment of the Chern character to the analytic context and specializes to Bloch's earlier description of the semiregularity map for locally complete intersections as well as to the infinitesimal Abel–Jacobi map for submanifolds.

The colored Jones polynomial is a function $J_K:{\mathbb{N}}\longrightarrow{\mathbb{Z}}[t,t^{-1}]$ associated with a knot $K$ in 3-space. We will show that for an alternating knot $K$ the absolute values of the first and the last three leading coefficients of $J_K(n)$ are independent of $n$ when $n$ is sufficiently large. Computation of sample knots indicates that this should be true for any fixed leading coefficient of the colored Jones polynomial for alternating knots. As a corollary we get a volume-ish theorem for the colored Jones polynomial.

Fomin–Zelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph and cluster complex are independent of the choice of coefficients. We confirm these conjectures for all skew-symmetric cluster algebras.