We show that the set of real polynomials in two variables that are sums of three squares of rational functions is dense in the set of those that are positive semidefinite. We also prove that the set of real surfaces in $\mathbb{P}^{3}$ whose function field has level 2 is dense in the set of those that have no real points.

The space of n distinct points and adisjoint parametrized hyperplane in projective d-space up to projectivity – equivalently, configurations of n distinct points in affine d-space up to translation and homothety – has a beautiful compactification introduced by Chen, Gibney and Krashen. This variety, constructed inductively using the apparatus of Fulton–MacPherson configuration spaces, is a parameter space of certain pointed rational varieties whose dual intersection complex is a rooted tree. This generalizes $\overline M _{0,n}$ and shares many properties with it. In this paper, we prove that the normalization of the Chow quotient of (ℙd)n by the diagonal action of the subgroup of projectivities fixing a hyperplane, pointwise, is isomorphic to this Chen–Gibney–Krashen space Td, n. This is a non-reductive analogue of Kapranov's famous quotient construction of $\overline M _{0,n}$, and indeed as a special case we show that $\overline M _{0,n}$ is the Chow quotient of (ℙ1)n−1 by an action of 𝔾m ⋊ 𝔾a.

We rewrite in modern language a classical construction by W. E. Edge showing a pencil of sextic nodal curves admitting A5 as its group of automorphism. Next, we discuss some other aspects of this pencil, such as the associated fibration and its connection to the singularities of the moduli of six-dimensional abelian varieties.

In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) $L_{\infty }$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_{\infty }$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_{\infty }$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_{\infty }$ space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated $L_{\infty }$ space.

Gromov–Witten invariants have been constructed to be deformation invariant, but their behavior under other transformations is subtle. We show that logarithmic Gromov–Witten invariants are also invariant under appropriately defined logarithmic modifications.

We first show that every algebraic torus over any field, not necessarily split, can be realized as the special fiber of a semi-abelian scheme whose generic fiber is an absolutely simple abelian variety. Then we investigate which algebraic tori can be thus obtained, when we require the generic fiber of the semi-abelian scheme to carry non-trivial endomorphism structures.

We show that integral monodromy groups of Kloosterman $\ell$-adic sheaves of rank $n\geqslant 2$ on $\mathbb{G}_{m}/\mathbb{F}_{q}$ are as large as possible when the characteristic $\ell$ is large enough, depending only on the rank. This variant of Katz’s results over $\mathbb{C}$ was known by works of Gabber, Larsen, Nori and Hall under restrictions such as $\ell$ large enough depending on $\operatorname{char}(\mathbb{F}_{q})$ with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz’s ideas, in particular the classification of maximal subgroups of Aschbacher and Kleidman–Liebeck. These results will apply to study hyper-Kloosterman sums and their reductions in forthcoming work.

Let $\unicode[STIX]{x1D709}$ be a stable Chern character on $\mathbb{P}^{1}\times \mathbb{P}^{1}$, and let $M(\unicode[STIX]{x1D709})$ be the moduli space of Gieseker semistable sheaves on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ with Chern character $\unicode[STIX]{x1D709}$. In this paper, we provide an approach to computing the effective cone of $M(\unicode[STIX]{x1D709})$. We find Brill–Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of $M(\unicode[STIX]{x1D709})$ which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on $\mathbb{P}^{1}\times \mathbb{P}^{1}$.

We present a Langlands dual realization of the putative category of affine character sheaves. Namely, we calculate the categorical center and trace (also known as the Drinfeld center and trace, or categorical Hochschild cohomology and homology) of the affine Hecke category starting from its spectral presentation. The resulting categories comprise coherent sheaves on the commuting stack of local systems on the two-torus satisfying prescribed support conditions, in particular singular support conditions, which appear in recent advances in the geometric Langlands program. The key technical tools in our arguments are a new descent theory for coherent sheaves or ${\mathcal{D}}$-modules with prescribed singular support and the theory of integral transforms for coherent sheaves developed in the companion paper by Ben-Zvi et al. [Integral transforms for coherent sheaves, J. Eur. Math. Soc. (JEMS), to appear].

This paper concerns the classification of isogeny classes of $p$-divisible groups with saturated Newton polygons. Let $S$ be a normal Noetherian scheme in positive characteristic $p$ with a prime Weil divisor $D$. Let ${\mathcal{X}}$ be a $p$-divisible group over $S$ whose geometric fibers over $S\setminus D$ (resp. over $D$) have the same Newton polygon. Assume that the Newton polygon of ${\mathcal{X}}_{D}$ is saturated in that of ${\mathcal{X}}_{S\setminus D}$. Our main result (Corollary 1.1) says that ${\mathcal{X}}$ is isogenous to a $p$-divisible group over $S$ whose geometric fibers are all minimal. As an application, we give a geometric proof of the unpolarized analogue of Oort’s conjecture (Oort, J. Amer. Math. Soc. 17(2) (2004), 267–296; 6.9).

A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\text{GL}_{n}(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles $\mathbb{A}_{X}^{\bullet }$, we have an equivalence $\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$ between perfect complexes on $X$ and cartesian perfect complexes for $\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal $\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

The detection of the bifurcation set of polynomial mapping ℝn → ℝp, n ⩾ p, in more than two variables remains an unsolved problem. In this note we provide a solution for n = p + 1 ⩾ 3.

This is the first of three papers in which we give a moduli interpretation of the second flip in the log minimal model program for $\overline{M}_{g}$, replacing the locus of curves with a genus $2$ Weierstrass tail by a locus of curves with a ramphoid cusp. In this paper, for $\unicode[STIX]{x1D6FC}\in (2/3-\unicode[STIX]{x1D716},2/3+\unicode[STIX]{x1D716})$, we introduce new $\unicode[STIX]{x1D6FC}$-stability conditions for curves and prove that they are deformation open. This yields algebraic stacks $\overline{{\mathcal{M}}}_{g}(\unicode[STIX]{x1D6FC})$ related by open immersions $\overline{{\mathcal{M}}}_{g}(2/3+\unicode[STIX]{x1D716}){\hookrightarrow}\overline{{\mathcal{M}}}_{g}(2/3){\hookleftarrow}\overline{{\mathcal{M}}}_{g}(2/3-\unicode[STIX]{x1D716})$. We prove that around a curve $C$ corresponding to a closed point in $\overline{{\mathcal{M}}}_{g}(2/3)$, these open immersions are locally modeled by variation of geometric invariant theory for the action of $\text{Aut}(C)$ on the first-order deformation space of $C$.

We prove a general criterion for an algebraic stack to admit a good moduli space. This result may be considered as a generalization of the Keel–Mori theorem, which guarantees the existence of a coarse moduli space for a separated Deligne–Mumford stack. We apply this result to prove that the moduli stacks $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ parameterizing $\unicode[STIX]{x1D6FC}$-stable curves introduced in [J. Alper et al., Second flip in the Hassett–Keel program: a local description, Compositio Math. 153 (2017), 1547–1583] admit good moduli spaces.

We prove that the direct image complex for the $D$-twisted $\text{SL}_{n}$ Hitchin fibration is determined by its restriction to the elliptic locus, where the spectral curves are integral. The analogous result for $\text{GL}_{n}$ is due to Chaudouard and Laumon. Along the way, we prove that the Tate module of the relative Prym group scheme is polarizable, and we also prove $\unicode[STIX]{x1D6FF}$-regularity results for some auxiliary weak abelian fibrations.

We show that the moduli spaces of stable sheaves on projective schemes admit certain non-commutative structures, which we call quasi-NC structures, generalizing Kapranov’s NC structures. The completion of our quasi-NC structure at a closed point of the moduli space gives a pro-representable hull of the non-commutative deformation functor of the corresponding sheaf developed by Laudal, Eriksen, Segal and Efimov–Lunts–Orlov. We also show that the framed stable moduli spaces of sheaves have canonical NC structures.

By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface over an algebraically closed field, a variety whose geometric generic fiber is a projective homogeneous space admits a rational point if and only if the elementary obstruction vanishes.

We show that the Craighero–Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero–Gattazzo surface is the only explicit example of a smooth simply connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface we use an algebraic reduction mod $p$ technique and deformation theory.

Let $G$ be a semi-simple algebraic group over an algebraically closed field $k$, whose characteristic is positive and does not divide the order of the Weyl group of $G$, and let $\breve{G}$ be its Langlands dual group over $k$. Let $C$ be a smooth projective curve over $k$ of genus at least two. Denote by $\operatorname{Bun}_{G}$ the moduli stack of $G$-bundles on $C$ and $\operatorname{LocSys}_{\breve{G}}$ the moduli stack of $\breve{G}$-local systems on $C$. Let $D_{\operatorname{Bun}_{G}}$ be the sheaf of crystalline differential operators on $\operatorname{Bun}_{G}$. In this paper we construct an equivalence between the bounded derived category $D^{b}(\operatorname{QCoh}(\operatorname{LocSys}_{\breve{G}}^{0}))$ of quasi-coherent sheaves on some open subset $\operatorname{LocSys}_{\breve{G}}^{0}\subset \operatorname{LocSys}_{\breve{G}}$ and bounded derived category $D^{b}(D_{\operatorname{Bun}_{G}}^{0}\text{-}\text{mod})$ of modules over some localization $D_{\operatorname{Bun}_{G}}^{0}$ of $D_{\operatorname{Bun}_{G}}$. This generalizes the work of Bezrukavnikov and Braverman in the $\operatorname{GL}_{n}$ case.