Smooth hypersurfaces in abelian varieties over arithmetic rings

The strongest notion is k-jet ampleness; it implies k-very ampleness (cf. [BeSo2, Proposition 2.2]) which of course implies k-spannedness. For k = 0 or k = 1 all the three notions are equivalent and correspond to global generation resp. very ampleness. In this note we give criteria for k-jet ampleness of line bundles on abelian varieties. A naive way to obtain such a criterion is as follows: According to [BeSo2, Corollary 2.1] a tensor product of k very ample line bundles is always k-jet ample. Now on an abelian variety, by the generalization of Lefschetz’ classical theorem [LB,

Very ample line bundles on regular surfaces obtained by projection

Let k be a field finitely generated over ℚ and p a prime. The torsion conjecture (resp. p-primary torsion conjecture) for abelian varieties over k predicts that the k-rational torsion (resp. the p-primary k-rational torsion) of a d-dimensional abelian variety A over k should be bounded only in terms of k and d. These conjectures are only known for d=1. The p-primary case was proved by Y. Manin, in 1969; the general case was completed by L. Merel, in 1996, after a series of contributions by B. Mazur, S. Kamienny and others. Due to the fact that moduli of elliptic curves are 1-dimensional, the d=1 case of the torsion conjecture (resp. p-primary torsion conjecture) is closely related to the following. For any k-curve S and elliptic scheme E→S, the k-rational torsion (resp. the p-primary k-rational torsion) is uniformly bounded in the fibres E s , s∈S(k). In this paper, we extend this result in the p-primary case to arbitrary abelian schemes over curves. More precisely, we prove the following. Denote by Γ k the absolute Galois group of k. For an abelian variety A over k and a character $\chi:\Gamma_{k}\rightarrow\mathbb{Z}_{p}^{\ast}$ , define A[p ∞](χ) to be the module of p-primary torsion of $A(\overline{k})$ on which Γ k acts as χ-multiplication. Assume that χ does not appear as a subrepresentation of the p-adic representation associated with an abelian variety over k. Then A[p ∞](χ) is always finite, but the exponent of A[p ∞](χ) may depend on A, a priori. Our main result is about the uniform boundedness of A[p ∞](χ) when A varies in a 1-dimensional family. More precisely, if S is a curve over k and A is an abelian scheme over S, then there exists an integer N:=N(A,S,k,p,χ), such that A s [p ∞](χ)⊂A s [p N ] holds for any s∈S(k). This arithmetic result is obtained as a corollary of the following geometric result on the p-primary torsion of abelian varieties over function fields of curves, combined with Mordell’s conjecture (Faltings’ theorem). Let K be the function field of a curve over an algebraically closed field of characteristic 0 and A an abelian variety over K. Assume for simplicity that A contains no nontrivial isotrivial subvariety. Then, for any c≥0, there exists an integer N:=N(c,A,S,k,p)≥0 such that A[p ∞](K′)⊂A[p N ] for all finite extension K′/K with K′ of genus ≤c. A key ingredient of the proof of this geometric result is a certain result on the number of points on reduction modulo p n of p-adic analytic homogeneous spaces. Our uniform boundedness result when χ is the trivial character gives the uniform boundedness for the k-rational p-primary torsion in the fibres A s , s∈S(k) alluded to above. When χ is the p-adic cyclotomic character, together with certain descent methods, it also yields a proof of the 1-dimensional case of (a generalized variant of) the modular tower conjecture, which was, actually, the original motivation for this work. This is a conjecture arising from the regular inverse Galois problem, whose original form was posed by M. Fried in the early 1990s.

Let pi :mathcal {X}rightarrow M be a holomorphic fibration with compact fibers and L a relatively ample line bundle over mathcal {X}. We obtain the asymptotic of the curvature of L^2-metric and Qullien metric on the direct image bundle pi _*(L^kotimes K_{mathcal {X}/M}) up to the lower order terms than k^{n-1}, for large k. As an application we prove that the analytic torsion tau _k(bar{partial }) satisfies partial bar{partial }log (tau _k(bar{partial }))^2=o(k^{n-1}), where n is the dimension of fibers.

On an abelian scheme over a smooth curve over $\bar{\mathbb{Q}}$ a symmetric relatively ample line bundle defines a fiberwise Neron–Tate height. If the base curve is inside a projective space, we also have a height on its $\bar{\mathbb{Q}}$-points that serves as a measure of each fiber, an abelian variety. Silverman proved an asymptotic equality between these two heights on a curve in the abelian scheme. In this paper we prove an inequality between these heights on a subvariety of any dimension of the abelian scheme. As an application we prove the Geometric Bogomolov Conjecture for the function field of a curve defined over $\bar{\mathbb{Q}}$. Using Moriwaki's height we sketch how to extend our result when the base field of the curve has characteristic 0.

We prove two new results for Seshadri constants on surfaces of general type. Let X be a surface of general type. In the first part, inspired by Bauer and Szemberg (Manuscripta Math 126(2):167–175, 2008), we list the possible values for the multi-point Seshadri constant $$\varepsilon (K_X,x_1,x_2,\ldots ,x_r)$$ when it lies between 0 and 1/r, where $$K_X$$ is the canonical line bundle on X. In the second part, we assume X of the form $$C \times C$$ , where C is a general smooth curve of genus $$g \geqslant 2$$ . Given such X and an ample line bundle L on X with some conditions on it, we show that the global Seshadri constant of L is a rational number.

In this paper, we study pushforwards of log pluricanonical bundles on projective log canonical pairs $(Y,\Delta)$ over the complex numbers, partially answering a Fujita-type conjecture due to Popa and Schnell in the log canonical setting. We show two effective global generation results. First, when $Y$ surjects onto a projective variety, we show a quadratic bound for generic generation for twists by big and nef line bundles. Second, when $Y$ is fibered over a smooth projective variety, we show a linear bound for twists by ample line bundles. These results additionally give effective non-vanishing statements. We also prove an effective weak positivity statement for log pluricanonical bundles in this setting, which may be of independent interest. In each context we indicate over which loci positivity holds. Finally, using the description of such loci, we show an effective vanishing theorem for pushforwards of certain log-sheaves under smooth morphisms.

In this paper we prove a refined version of the canonical key formula for projective abelian schemes in the sense of Moret-Bailly (cf. Asterisque 129, 1985), we also extend this discussion to the context of Arakelov geometry. Precisely, let \({\pi: A \to S}\) be a projective abelian scheme over a locally noetherian scheme S with unit section \({e: S \to A}\) and let L be a symmetric, rigidified, relatively ample line bundle on A. Denote by ωA the determinant of the sheaf of differentials of π and by d the rank of the locally free sheaf π*L. In this paper, we shall prove the following results: (i). there is an isomorphism $${\rm det}(\pi_*L)^{\otimes 24} \cong (e^*\omega_A^\vee)^{\otimes 12d}$$ which is canonical in the sense that it can be chosen to be functorial, namely it is compatible with arbitrary base-change; (ii). if the generic fibre of S is separated and smooth, then there exist a positive integer m and canonical metrics on L and on ωA such that there exists an isometry $${\rm det}(\pi_*\overline{L})^{\otimes 2m} \cong (e^*\overline{\omega}_A^\vee)^{\otimes md}$$ which is canonical in the sense of (i). Here the constant m only depends on g, d and is independent of L.

We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields of characteristic p . We prove the prime-to- p part conditionally on the finiteness of the p -primary part of the Tate-Shafarevich group or the equality of the analytic and the algebraic rank. If the base is a product of curves, Abelian varieties and K3 surfaces, we prove the prime-to- p part of the conjecture for constant or isoconstant Abelian schemes, in particular the prime-to- p part for (1) relative elliptic curves with good reduction or (2) Abelian schemes with constant isomorphism type of \mathscr{A}[p] or (3) Abelian schemes with supersingular generic fibre, and the full conjecture for relative elliptic curves with good reduction over curves and for constant Abelian schemes over arbitrary bases. We also reduce the conjecture to the case of surfaces as the basis.

We describe linear algebra algorithms for doing arithmetic on an abelian variety which is dual to a given abelian variety. The ideas are inspired by Khuri-Makdisi’s algorithms for Jacobians of curves. Let \(\chi _{0}\) be the Euler characteristic of the line bundle associated with an ample divisor H on an abelian variety A. The Hilbert scheme of effective divisors D such that \({\mathscr {O}}(D)\) has Hilbert polynomial \((1+t)^g\chi _{0}\) is a projective bundle (with fibres \({\mathbb {P}}^{\chi _{0}-1}\)) over the dual abelian variety \(\widehat{A}\) via the Abel–Jacobi map. This Hilbert scheme can be embedded in a Grassmannian, so that points on it (and hence, via the above-mentioned Abel–Jacobi map, points on \(\widehat{A}\)) can be represented by matrices. Arithmetic on \(\widehat{A}\) can be worked out by using linear algebra algorithms on the representing matrices.

In this paper we study the moduli space A0 Fp of polarized abelian varieties of dimensiongin positive characteristic. We construct a stratification of this space. The strata are indexed by isomorphism classes of group schemes killed byp;a polarized abelian variety (X, A) has its moduli point in a certain stratum if X [p] belongs to the isomorphism class given by a certain discrete invariant. We define these invariants by a numerical property of a filtration ofN= X [p].Passing from one stratum to a stratum in its boundary feels like “degenerating the p-structure”. The fact that these strata are all quasi-affine allows us to keep going in this process until we arrive at the unique zero-dimensional stratum, the superspecial locus. One can formulate this idea by saying that the ordinary locus has several “boundaries”, one where the abelian variety degenerates, one where the p-structure “becomes more special” (and an analogous idea for all non-zero-dimensional strata). This phenomenon, non-present in this form in characteristic zero, but available and powerful in positive characteristic, is ex-pected to have many applications.KeywordsModulus SpaceAbelian VarietyGroup SchemeNewton PolygonAbelian SchemeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we prove the results announced in [L-R] for G = SLn. Let G be a simple algebraic group over the base field k. Let M be a maximal torus in G and B, a Borel subgroup, B ⊃ M. Let W be the Weyl group of G. For w ∈ W, let X(w) = \(\overline {BwB} \) (mod B) be the Schubert variety in G/B associated to w. Let L be an ample line bundle on G/B. We shall denote the restriction of L to X(w) also by just L. Let k[X(w)] = ⊙ H0(X(w),L). In this paper, we construct an algebra kq [X(w)] over k(q), where q is a parameter taking values in k*, as a quantization of k[X(w)], G being SLn. The algebra k q [SL n ]: Let G = SLn. Let T = (tij), 1 ≤ i, j ≤ n. Let Let $$R = \sum\limits_{\mathop {i \ne j}\limits_{i,j = 1} }^n {{e_{ii}}} { \otimes _{jj}} + q\sum\limits_{i \ne i}^n {{e_{ii}}} \otimes {e_{ii}} + \left( {q - {q^{ - 1}}} \right)\sum\limits_{1j < in} {{e_{ij}}} \otimes {e_{ji}}$$ (here, eij’s are the elementary matrices). Let A(R) be the associative algebra (with 1) generated by {tij, 1 ≤ i, j ≤ n}, the relations being given by RT1T2 = T2T1R, where T1 = T ⊗ Id, T2 = Id ⊗ T (cf. [F-R-T]). Then A(R) gives a quantization of k[Mn], Mn being the space of n × n matrices and k[Mn], the coordinate ring of Mn. Now A(R) has a bialgebra structure, then comultiplication being given by Δ: A(R) → A(R) ⊗ A(R), \(\Delta = \left( {{t_{ij}}} \right) = \sum\limits_{k = 1}^n {{t_{ik}}} \otimes {t_{kj}}\) In the sequel, we shall denote A(R) by kq [Mn]. Let $$D = \sum\limits_{\sigma \in {s_n}}^n {{t_{ik}}} {\left( { - q} \right)^{ - l{{\left( \sigma \right)}_t}}}1\sigma {\left( 1 \right)^t}2\sigma {\left( 2 \right)^{...}}n\sigma \left( n \right)$$ . we shall refer to D as the q-determinant (or the quantum determinant) of (tij).

Let $H$ and $H'$ be two ample line bundles over a nonsingular projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r=2,c_1,c_2)$. In a moduli-theoretic way that comes from elementary transforms, we connect $M(H)$ and $M(H')$ by a sequence of blowing-ups when walls separating $H$ and $H'$ are not necessarily good. As an application, we also consider the polarization change problem of Donaldson polynomials.

Semistability and Hilbert–Kunz multiplicities for curves

Let ϵ be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of ϵ whose zero locus Z is a smooth subvariety of dimension n-r ≥ 2 of X. Let H be an ample line bundle on X such that the restriction HZ of H to Z is very ample. Triplets (X, ϵ, H) with g(Z, HZ) = 3 are classified, where g(Z, HZ) is the sectional genus of (Z, HZ).