Our main result is the description of generators of the total coordinate ring of the blow-up of $\mathbb{P}^n$ in any number of points that lie on a rational normal curve. As a corollary we show that the algebra of invariants of the action of a two-dimensional vector group introduced by Nagata is finitely generated by certain explicit determinants. We also prove the finite generation of the algebras of invariants of actions of vector groups related to T-shaped Dynkin diagrams introduced by Mukai.

A fundamental object in the theory of arithmetic surfaces is the Green's function associated to the canonical metric. Previous expressions for the canonical Green's function have relied on general functional analysis or, when using specific properties of the canonical metric, the classical Riemann theta function. In this article, we derive a new identity for the canonical Green's function involving the hyperbolic heat kernel. As an application of our results, we obtain bounds for the canonical Green's function through covers and for families of modular curves.

We classify all Gorenstein Fano threefolds with at worst canonical singularities for which the anticanonical system |−K| has a nonempty base locus.

The number of flags in a complete fan, or more generally in an Eulerian poset, is encoded in the cd-index. We prove the non-negativity of the cd-index for complete fans, regular CW-spheres and Gorenstein* posets.

Generalizing a conjecture of Deligne, Kontsevich proposed that there should be a notion of Hochschild cohomology of algebras over the little cube operad (or its chain complex) which in a natural way generalizes Hochschild cohomology of associative algebras. He moreover conjectured that the Hochschild cohomology, in this new sense, of an algebra over the little k-cube operad is an algebra over the little (k + 1)-cube operad. In this paper, we precisely state and prove this conjecture.

Using multigraded Castelnuovo–Mumford regularity, we study the equations defining a projective embedding of a variety $X$. Given globally generated line bundles $B_{1}, \dotsc, B_{\ell}$ on $X$ and $m_{1}, \dotsc, m_{\ell} \in \mathbb{N}$, consider the line bundle $L := B_{1}^{m_{1}} \otimes \dotsb \otimes B_{\ell}^{m_{\ell}}$. We give conditions on the $m_{i}$ which guarantee that the ideal of $X$ in $\mathbb{P}(H^{0}(X,L)^{*})$ is generated by quadrics and that the first $p$ syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.

We investigate the bounded derived category of coherent sheaves on irreducible singular projective curves of arithmetic genus one. A description of the group of exact auto-equivalences and the set of all $t$-structures of this category is given. We describe the moduli space of stability conditions, obtain a complete classification of all spherical objects in this category and show that the group of exact auto-equivalences acts transitively on them. Harder–Narasimhan filtrations in the sense of Bridgeland are used as our main technical tool.

We propose a procedure to construct new smooth CR-manifolds whose local stability groups, equipped with their natural topologies, are subgroups of certain (finite-dimensional) Lie groups but not Lie groups themselves.

King's conjecture states that on every smooth complete toric variety $X$ there exists a strongly exceptional collection which generates the bounded derived category of $X$ and which consists of line bundles. We give a counterexample to this conjecture. This example is just the Hirzebruch surface $\mathbb{F}_2$ iteratively blown up three times, and we show by explicit computation of cohomology vanishing that there exist no strongly exceptional sequences of length 7 which consist of line bundles.

We define nef line bundles ${\mathcal L}_r$ on a projective variety X with the property that, for a curve $C \subset X$, the intersection ${\mathcal L}_r.C$ is zero, if and only if the restriction morphism ${\rm Hom}(\pi_1(X),{\rm U}(r)) \to {\rm Hom}(\pi_1(C),{\rm U}(r))$ has finite image up to conjugation. This yields a rational morphism $\smash[b]{\xymatrix{X \ar@{-->}[r]^-{{\rm alb}_r} & {\rm Alb}_r(X)}}$ contracting those curves C with ${\mathcal L}_r.C=0$. For $r=1$ this is the Stein factorization of the Albanese morphism.

Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of rational functions (up to automorphism of the target) on the projective line with ramification to order ei at general points Pi, in the case that all ei are less than the characteristic. Unlike the case of characteristic 0, the answer is not given by Schubert calculus, nor is the number of maps always finite for distinct Pi, even in the tamely ramified case. However, finiteness for general Pi, obtained by exploiting the relationship to branched covers, is a key part of the argument.

We apply the methods developed in our previous work to study holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds. In particular, we extend and strengthen certain results of Gromov, Henkin and Shubin on holomorphic $L^{2}$ functions on coverings of pseudoconvex manifolds in the case of coverings of Stein manifolds.

We prove that holonomic arithmetical ${\mathcal{D}}$-modules over curves have finite fibers. We also define L-functions associated with arithmetical ${\mathcal{D}}$-modules and, when the scheme is a curve, we show a cohomological formula. Furthermore, we prove that F-isocrystals over curves are holonomic.

We conjecture an equivalence between the Gromov–Witten theory of 3-folds and the holomorphic Chern–Simons theory of Donaldson and Thomas. For Calabi–Yau 3-folds, the equivalence is defined by the change of variables $e^{iu}=-q$, where $u$ is the genus parameter of Gromov–Witten theory and $q$ is the Euler characteristic parameter of Donaldson–Thomas theory. The conjecture is proven for local Calabi–Yau toric surfaces.

Let R be a d-dimensional regular local ring of characteristic p > 0 with maximal ideal $\mathfrak m$, let I be an ideal of R and let A = R/I. We describe some properties of the local cohomology module HiI(R), in particular its vanishing, in terms of the Frobenius action on the local cohomology module $H^{d-i}_{\mathfrak m}(A)$.

We study the Chow group of zero-dimensional cycles for projective homogeneous varieties of semisimple algebraic groups. We show that in many cases this group has no torsion.

We characterise the class of finite solvable groups by two-variable identities in a way similar to the characterisation of finite nilpotent groups by Engel identities. Let $u_1=x^{-2}y\min x$, and $u_{n+1} = [xu_nx\min,yu_ny\min]$. The main result states that a finite group G is solvable if and only if for some n the identity $u_n(x,y)\equiv 1$ holds in G. We also develop a new method to study equations in the Suzuki groups. We believe that, in addition to the main result, the method of proof is of independent interest: it involves surprisingly diverse and deep methods from algebraic and arithmetic geometry, topology, group theory, and computer algebra (SINGULAR and MAGMA).

Let $B_n$ denote the classical braid group on $n$ strands and let the mixed braid group$B_{m,n}$ be the subgroup of $B_{m+n}$ comprising braids for which the first $m$ strands form the identity braid. Let $B_{m,\infty}=\bigcup_nB_{m,n}$. We describe explicit algebraic moves on $B_{m,\infty}$ such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented three-manifold. The moves depend on a fixed link representing the manifold in $S^3$. More precisely, for link complements the moves are the two familiar moves of the classical Markov equivalence together with ‘twisted’ conjugation by certain loops $a_i$. This means premultiplication by $a_i^{-1}$ and postmultiplication by a ‘combed’ version of $a_i$. For closed three-manifolds there is an additional set of ‘combed’ band moves that correspond to sliding moves over the surgery link. The main tool in the proofs is the one-move Markov theorem using $L$-moves (adding in-box crossings). The resulting algebraic classification is a direct extension of the classical Markov theorem that classifies links in $S^3$ up to isotopy, and potentially leads to powerful new link invariants, which have been explored in special cases by the first author. It also provides a controlled range of isotopy moves, useful for studying skein modules of three-manifolds.

Suppose that R is a two-dimensional normal standard-graded domain over a finite field. We prove that there exists a uniform Frobenius test exponent b for the class of homogeneous ideals in R generated by at most n elements. This means that for every ideal I in this class we have that $f^{p^b}\in I^{[p^b]}$ if and only if $f\in I^{\rm F}$. This gives in particular a finite test for the Frobenius closure. On the other hand we show that there is no uniform bound for Frobenius test exponent for all homogeneous ideals independent of the number of generators. Under similar assumptions we prove also the existence of a bound for tight closure test ideal exponents for ideals generated by at most n elements.

We discuss the Gromov–Witten/Donaldson–Thomas correspondence for 3-folds in both the absolute and relative cases. Descendents in Gromov–Witten theory are conjectured to be equivalent to Chern characters of the universal sheaf in Donaldson–Thomas theory. Relative constraints in Gromov–Witten theory are conjectured to correspond in Donaldson–Thomas theory to cohomology classes of the Hilbert scheme of points of the relative divisor. Independent of the conjectural framework, we prove degree 0 formulas for the absolute and relative Donaldson–Thomas theories of toric varieties.