Manuscripta Mathematica最新文献

Some classes of cubic fourfolds are birational to fibrations over ({mathbb {P}}^2), where the fibers are rational surfaces. This is the case for cubics containing a plane (resp. an elliptic ruled surface), where the fibers are quadric surfaces (resp. del Pezzo sextic surfaces). It is known that the rationality of these cubic hypersurfaces is related to the rationality of these surfaces over the function field of ({mathbb {P}}^2) and to the existence of rational (multi)sections of the fibrations. We study, in the moduli space of cubic fourfolds, the intersection of the divisor ({mathcal {C}}_{8}) (resp. ({mathcal {C}}_{18})) with ({mathcal {C}}_{14}), ({mathcal {C}}_{26}) and ({mathcal {C}}_{38}), whose elements are known to be rational cubic fourfolds. We provide descriptions of the irreducible components of these intersections and give new explicit examples of rational cubics fibered in (quartic, quintic) del Pezzo surfaces or in quadric surfaces over ({mathbb {P}}^2). We also investigate the existence of rational sections for these fibrations. Under some mild assumptions on the singularities of the fibers, these properties can be translated in terms of Brauer classes on certain surfaces.

We generalize the work of Lindenstrauss and Venkatesh establishing Weyl’s Law for cusp forms from the spherical spectrum to arbitrary archimedean type. Weyl’s law for the spherical spectrum gives an asymptotic formula for the number of cusp forms that are bi-(K_{infty }) invariant in terms of eigenvalue T of the Laplacian. We prove that an analogous asymptotic holds for cusp forms with archimedean type (tau ), where the main term is multiplied by (dim {tau }). While in the spherical case, the surjectivity of the Satake Map was used, in the more general case that is not available and we use Arthur’s Paley–Wiener theorem and multipliers.

We study line bundles on smooth toric Deligne-Mumford stacks ({mathbb {P}}_{mathbf {Sigma }}) of arbitrary dimension. We give a sufficient condition for when infinitely many line bundles on ({mathbb {P}}_{mathbf {Sigma }}) have trivial cohomology. In dimension three, this sufficient condition is also a necessary condition under the technical assumption that (mathbf {Sigma }) has no more than one pair of collinear rays.

Let X be an adic space locally of finite type over a complete non-archimedean field k, and denote ({textbf {Cov}}_{X}^{textrm{oc}}) (resp. ({textbf {Cov}}_{X}^{textrm{adm}})) the category of étale coverings of X that are locally for the Berkovich overconvergent topology (resp. for the admissible topology) disjoint union of finite étale coverings. There is a natural inclusion ({textbf {Cov}}_{X}^{textrm{oc}}subseteq {textbf {Cov}}_{X}^{textrm{adm}}). Whether or not this inclusion is strict is a question initially asked by de Jong. Some partial answers have been given in the recents works of Achinger, Lara and Youcis in the finite or equal characteristic 0 cases. The present note shows that this inclusion can be strict when k is of mixed characteristic (0, p) and p-closed. As a consequence, the natural morphism of Noohi groups (pi _1^{mathrm {dJ, , adm}}(mathcal {C}, overline{x})rightarrow pi _1^{mathrm {dJ, ,oc}}(mathcal {C},overline{x}) ) is not an isomorphism in general.

We compute the integral Grothendieck rings of the moduli stacks, ({{mathcal {M}}}_2), (overline{{{mathcal {M}}}}_2) of smooth and stable curves of genus two respectively. We compute ({{,textrm{K},}}_0({{mathcal {M}}}_2)) by using the presentation of ({{mathcal {M}}}_2) as a global quotient stack given by Vistoli (Invent Math 131(3):635–644, 1998). To compute the Grothendieck ring ({{,textrm{K},}}_0(overline{{{mathcal {M}}}}_2)) we decompose (overline{{{mathcal {M}}}}_2) as (Delta _1) and its complement (overline{{{mathcal {M}}}}_2 setminus Delta _1) and use their presentations as quotient stacks given by Larson (Algebr Geom 8 (3):286–318, 2021) to compute the Grothendieck rings. We show that they are torsion-free and this, together with the Riemann–Roch isomorphism allows us to ultimately give a presentation for the integral Grothendieck ring ({{,textrm{K},}}_0(overline{{{mathcal {M}}}}_2)).

We show that shrinking Kähler-Ricci solitons over a compact Kähler manifold are gradient shrinking Kähler-Ricci solitons. The proof relies on a remarkable identity on the kernels of a real and a complex elliptic operator proved in our solution of the variational stability problem for gradient shrinking Kähler-Ricci solitons in Pali (Complex Manifolds 3(1):41–144, 2016).

For an elliptic surface (pi :Xrightarrow mathbb {P}^1) defined over a number field K, a theorem of Silverman shows that for all but finitely many fibres above K-rational points, the resulting elliptic curve over K has Mordell-Weil rank at least as large as the rank of the group of sections of (pi ). When X is a K3 surface with two distinct elliptic fibrations, we show that the set of K-rational points of (mathbb {P}^1) for which this rank inequality is strict, is not a thin set, under certain hypothesis on the fibrations. Our results provide one of the first cases of this phenomenon beyond that of rational elliptic surfaces.

Let (mathcal {I}_{d,g,r}) be the union of irreducible components of the Hilbert scheme whose general points represent smooth, irreducible, non-degenerate curves of degree d and genus g in (mathbb {P}^r). Using a family of curves found on ruled surfaces over smooth curves of genus (gamma ), we show that for (gamma ge 7) and (g ge 6 gamma + 5), the scheme (mathcal {I}_{2g-4gamma + 1, g, g - 3gamma + 1}) acquires a non-reduced component (mathcal {D}^{prime }) such that ({text {dim}}T_{[X^{prime }]} mathcal {D}^{prime } = {text {dim}}mathcal {D}^{prime } + 1) for a general point ([X^{prime }] in mathcal {D}^{prime }).

In this paper, we describe the structure of the negative part of a Zariski decomposition of (K_X+K_{{{mathcal {F}}}}) for a relatively minimal foliation ((X,{{mathcal {F}}})) whenever (K_X+K_{{{mathcal {F}}}}) is pseudoeffective.

Let (ell ) be a prime number different from the residue characteristic of a non-archimedean local field F. We give formulations of (ell )-adic local Langlands correspondences for connected reductive algebraic groups over F, which we conjecture to be independent of a choice of an isomorphism between the (ell )-adic coefficient field and the complex number field.