We determine the Brauer group of the Deligne-Mumford stack $\mathscr{Y}_0(2)$, the moduli space of elliptic curves with a marked $2$-torsion subgroup over bases of arithmetic interest. Antieau and Meier determine the Brauer group for $\mathscr{M}_{1,1}$, the moduli stack of elliptic curves by exploiting the fact it is covered by the Legendre family and using the Hochschild-Serre spectral sequence. Over an algebraically closed field, Shin uses the coarse space map to determine the Brauer group of $\mathscr{M}_{1,1}$. We combine techniques from both papers to determine the Brauer group of $\mathscr{Y}_0(2)$.
Given a log Calabi-Yau surface (Y,D), Bousseau has constructed a quantization of the mirror algebra of this pair. We give a formula for structure constants of this quantization in terms of higher-genus descendant logarithmic Gromov-Witten invariants of (Y, D). Our result generalises the weak Frobenius structure conjecture for surfaces to the q-refined setting, and is proved by relating these invariants to counts of quantum broken lines in the associated quantum scattering diagram.
We extend the notions of Hochschild and cyclic homology to morphisms from algebraic spaces to algebraic stacks. Using this, we obtain generalizations to log schemes in the sense of Fontaine and Illusie of these homology theories.
It is well known that a non-singular real plane projective curve of degree five with five connected components is separating if and only if its ovals are in non-convex position. In this article, this property is set into a different context and generalised to all real plane separating (M-2)-curves.
In this article, we present two structural results about the Renaudineau-Shaw spectral sequence that computes the cohomology of T-hypersurfaces. The first is a Poincar & eacute; duality satisfied by all its pages of positive index. The second is a vanishing criterion. It reformulates the vanishing of the boundary operators of the spectral sequence as the injectivity of some morphisms induced in cohomology by the inclusion of the T-hypersurface in its surrounding toric variety. It implies that the Renaudineau-Shaw spectral sequence of a T-hypersurface degenerates at the second page if and only if the T-hypersurface satisfies a real version of the Lefschetz hyperplane section theorem.
We generalize techniques by Coskun, Riedl, and Yeong, and obtain an almost optimal bound on the degree for the algebraic hyperbolicity of very general hypersurfaces in homogeneous varieties. As examples, we work out the cases of very general hypersurfaces in products of Grassmannians, orthogonal and symplectic Grassmannians, and flag varieties.
Enriques manifolds are non-simply connected manifolds whose universal cover is irreducible holomorphic symplectic, and as such they are natural generalizations of Enriques surfaces. The goal of this note is to prove the Morrison-Kawamata cone conjecture for very general Enriques manifolds when the degree of the cover is prime. The proof uses the analogous result (established by Amerik-Verbitsky) for their universal cover. We also verify the conjecture for a very general Enriques manifold which is deformation equivalent to one of the known examples.
In this paper, we establish formulas for computing genus-0 Gromov-Witten and Welschinger invariants of some del Pezzo varieties of dimension three by comparing to that of dimension two. These formulas are generalizations of that given in three-dimensional projective space by E. Brugallé and P. Georgieva in 2016.
For any rigid analytic group variety $G$ over a non-archimedean field $K$ over $\mathbb Q_p$, we study $G$-torsors on adic spaces over $K$ in the $v$-topology. Our main result is that on perfectoid spaces, $G$-torsors in the \'etale and $v$-topology are equivalent. This generalises the known cases of $G=\mathbb G_a$ and $G=\mathrm{GL}_n$ due to Scholze and Kedlaya--Liu. On a general adic space $X$ over $K$, where there can be more $v$-topological $G$-torsors than \'etale ones, we show that for any open subgroup $U\subseteq G$, any $G$-torsor on $X_v$ admits a reduction of structure group to $U$ \'etale-locally on $X$. This has applications in the context of the $p$-adic Simpson correspondence: For example, we use it to show that on any adic space, generalised $\mathbb Q_p$-representations are equivalent to $v$-vector bundles.
We determine the all-genus Hodge-Gromov-Witten theory of a smooth hypersurface in weighted projective space defined by a chain or loop polynomial. In particular, we obtain the first genus zero computation of Gromov-Witten invariants for hypersurfaces in non-Gorenstein ambiant spaces, where the convexity property fails. We extend it to any weighted projective hypersurface defined by an invertible polynomial.
In this note, we give a new proof of Voisin's theorem on canonical syzygies for generic curves of odd genus.
Multi-scale differentials were constructed by M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, and M. M & ouml;ller, from the viewpoint of flat and complex geometry, for the purpose of compactifying moduli spaces of curves together with a differential with prescribed orders of zeros and poles. Logarithmic differentials were constructed by S. Marcus and J. Wise, as a generalization of stable rubber maps from Gromov-Witten theory. Modulo the global residue condition that isolates the main components of the compactification, we show that these two kinds of differentials are equivalent, and establish an isomorphism of their (coarse) moduli stacks. Moreover, we describe the rubber and multi-scale spaces as an explicit blowup of the moduli space of stable pointed rational curves in the case of genus zero, and as a global blowup of the incidence variety compactification for arbitrary genera, which implies their projectivity. We also propose a refined double ramification cycle formula in the twisted Hodge bundle which interacts with the universal line bundle class.
We extend results of Looijenga-Lunts and Verbitsky and show that the total Lie algebragfor the intersection cohomology of a primitive symplectic varietyXwith isolated singularities isisomorphic togsoIH2(X,Q),QX h,whereQXis the intersection Beauville-Bogomolov-Fujiki form andhis a hyperbolic plane. Thisgives a new,algebraicproof for irreducible holomorphic symplectic manifolds which does not relyon the hyperk & auml;hler metric.Along the way, we study the structure ofIH & lowast;(X,Q)as ag-representation-with particular em-phasis on the Verbitsky component, multidimensional Kuga-Satake constructions, and Mumford-Tate algebras-and give some immediate applications concerning theP=Wconjecture for prim-itive symplectic varieties.
We prove that the ideal of relations in the (equivariant) quantum K-ring of a homogeneous space is generated by quantizations of each of the generators of the ideal in the classical (equivariant) K-ring. This extends to quantum K-theory a result of Siebert and Tian in quantum cohomology. We illustrate this technique in the case of the quantum K-ring of partial flag manifolds, using a set of quantum K-Whitney relations conjectured by the authors, and recently proved by Huq-Kuruvilla.
We use dualities of quiver moduli induced by reflection functors to describe generating series of motives of Kronecker moduli spaces of central slope as solutions of algebraic and q-difference equations.
An action of a complex reductive group G on a smooth projective variety X is regular when all regular unipotent elements in G act with finitely many fixed points. Then the complex G-equivariant cohomology ring of X is isomorphic to the coordinate ring of a certain regular fixed point scheme. Examples include partial flag varieties, smooth Schubert varieties and Bott-Samelson varieties. We also show that a more general version of the fixed point scheme allows a generalisation to GKM spaces, such as toric varieties.
In this note, we give a new proof of Voisin's theorem on Green's conjecture for generic curves of odd genus resembling the first two sections of "Universal Secant Bundles and Syzygies of Canonical Curves" by the author, and so avoiding the need for difficult computations.
Let L be a holomorphic line bundle on a hyperk & auml;hler manifold M, with c1(L) nef and not big. The SYZ conjecture predicts that L is semiample. We prove that this is true, assuming that (M, L) has a deformation (M ', L ') with L ' semiample. We introduce a version of the Teichm & uuml;ller space that parametrizes pairs (M, L) up to isotopy. We prove a version of the global Torelli theorem for such Teichm & uuml;ller spaces and use it to deduce the deformation invariance of semiampleness.
The quotient variety associated to a permutation representation of a finite group has only canonical singularities in arbitrary characteristic. Moreover, the log pair associated to such a representation is Kawamata log terminal except in characteristic two, and log canonical in arbitrary characteristic.
We study the relationship between solutions to better-behaved GKZ hypergeometric systems near different large radius limit points, and their geometric counterparts given by the K-groups of the associated toric Deligne-Mumford stacks. We prove that the K-theoretic Fourier-Mukai transforms associated to toric wall-crossing coincide with analytic continuation transformations of Gamma series solutions to the better-behaved GKZ systems, which settles a conjecture of Borisov and Horja.
