Showing papers in "Journal of Algebraic Geometry in 2009"

Comparing powers and symbolic powers of ideals

Abstract: We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results on the structure of the set of pairs $(r,m)$ such that $I^{(m)}\subseteq I^r$. As corollaries, we show that $I^2$ contains $I^{(3)}$ whenever $S$ is a finite generic set of points in ${\bf P}^2$ (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.

Milnor -theory of local rings with finite residue fields

Abstract: We propose a definition of improved Milnor K-groups of local rings with finite residue fields, such that the improved Milnor Ksheaf in the Zariski topology is a universal extension of the naive Milnor K-sheaf with a certain transfer map for etale extensions of local rings. The main theorem states that the improved Milnor K-ring is generated by elements of degree one.

Arrangements of curves and algebraic surfaces

Abstract: We prove a strong relation between Chern and log Chern invariants of algebraic surfaces. For a given arrangement of curves, we find nonsingular projective surfaces with Chern ratio arbitrarily close to the log Chern ratio of the log surface defined by the arrangement. Our method is based on sequences of random p-th root covers, which exploit a certain large scale behavior of Dedekind sums and lengths of continued fractions. We show that randomness is necessary for our asymptotic result, providing another instance of "randomness implies optimal". As an application over the complex numbers, we construct nonsingular simply connected projective surfaces of general type with large Chern ratio. In particular, we improve the Persson-Peters-Xiao record for Chern ratios of such surfaces.

Root numbers, Selmer groups, and non-commutative Iwasawa theory

Abstract: Let E be an elliptic curve over a number field F, and let F ∞ be a Galois extension of F whose Galois group G is a p-adic Lie group. The aim of the present paper is to provide some evidence that, in accordance with the main conjectures of Iwasawa theory, there is a close connection between the action of the Selmer group of E over F ∞ , and the global root numbers attached to the twists of the complex L-function of E by Artin representations of G.

Eigencone, saturation and horn problems for symplectic and odd orthogonal groups

Abstract: In this paper we consider the eigenvalue problem, intersection theory of homogeneous spaces (in particular, the Horn problem) and the saturation problem for the symplectic and odd orthogonal groups. The classical embeddings of these groups in the special linear groups play an important role. We deduce properties for these classical groups from the known properties for the special linear groups.

Non-archimedean analytification of algebraic spaces

Abstract: It is now a classical result that an algebraic space locally of finite type over $\mathbf{C}$ is analytifiable if and only if it is locally separated. In this paper we study non-archimedean analytifications of algebraic spaces. We construct a quotient for any etale non-archimedean analytic equivalence relation whose diagonal is a closed immersion, and deduce that any separated algebraic space locally of finite type over any non-archimedean field $k$ is analytifiable in both the category of rigid spaces and the category of analytic spaces over $k$. Also, though local separatedness remains a necessary condition for analytifiability in either of these categories, we present many surprising examples of non-analytifiable locally separated smooth algebraic spaces over $k$ that can even be defined over the prime field.

Inducing stability conditions

Abstract: We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a smooth projective variety we prove that the subset of invariant stability conditions embeds as a closed submanifold into the stability manifold of the equivariant derived category. As an application we examine stability conditions on Kummer and Enriques surfaces and we improve the derived version of the Torelli Theorem for the latter surfaces already present in the litterature. We also study the relationship between stability conditions on projective spaces and those on their canonical bundles.

On the geometry of the moduli space of spin curves

Abstract: We determine the smooth locus and the locus of canonical singularities in the Cornalba compactification \bar S_g of the moduli space S_g of spin curves, i.e., smooth curves of genus g with a theta characteristic. Moreover, the following lifting result for pluricanonical forms is proved: Every pluricanonical form on the smooth locus of \bar S_g extends holomorphically to a desingularisation of \bar S_g.

Rational maps between moduli spaces of curves and Gieseker-Petri divisors

Abstract: is injective. The theorem, conjectured by Petri and proved by Gieseker [G] (see [EH3] for a much simplified proof), lies at the cornerstone of the theory of algebraic curves. It implies that the variety Gd(C) = {(L, V ) : L ∈ Pic (C), V ∈ G(r + 1,H0(L))} of linear series of degree d and dimension r is smooth and of expected dimension ρ(g, r, d) := g− (r+1)(g−d+ r) and that the forgetful mapGd(C) → W r d (C) is a rational resolution of singularities (see [ACGH] for many other applications). It is an old open problem to describe the locus GPg ⊂ Mg consisting of curves [C] ∈ Mg such that there exists a line bundle L on C for which the Gieseker-Petri theorem fails. Obviously GPg breaks up into irreducible components depending on the numerical types of linear series. For fixed integers d, r ≥ 1 such that g − d + r ≥ 2, we define the locus GPg,d consisting of curves [C] ∈ Mg such that there exist a pair of linear series (L, V ) ∈ G r d(C) and (KC ⊗ L ∨,W ) ∈ G 2g−2−d (C) for which the multiplication map μ0(V,W ) : V ⊗W → H (C,KC )

Effective Iitaka fibrations

Abstract: For every n-dimensional projective manifold X of Kodaira dimension 2 we show that Φ|MKX | is birational to an Iitaka fibration for a computable positive integer M = M(b, Bn−2), where b > 0 is minimal with |bKF | 6 = ∅ for a general fibre F of an Iitaka fibration of X, and where Bn−2 is the Betti number of a smooth model of the canonical Z/(b)-cover of the (n − 2)-fold F . In particular, M is a universal constant if the dimension n ≤ 4. Building up on the work of H. Tsuji, C.D. Hacon and J. McKernan in [HM] and independently S. Takayama in [Ta] have shown the existence of a constant rn such that Φ|mKX | is a birational map for every m ≥ rn and for every complex projective n-fold X of general type. If the Kodaira dimension κ = κ(X) 0 | |bKF | 6 = ∅}, and Bn−κ to be the (n − κ)-th Betti number of a nonsingular model of the Z/(b)-cover of F , obtained by taking the b-th root out of the unique member in |bKF |, or as we will say, the middle Betti number of the canonical covering of F . Question 0.1. Is there a constant M := M(n, b, Bn−κ) such that Φ|MKX | is (birational to) an Iitaka fibration f : X → Y for all projective n-folds X of Kodaira dimension κ? Assume that for all s ≤ n there exists an effective constant a(s) such that for every projective s-fold V one has |a(s)KV | 6= ∅ and such that the dimension of |a(s)KV | is at least one if κ(V ) > 0. Then J. Kollar gives in [Ko86, Th 4.6] a formula for the constant M in 0.1 in terms of a(s) and n. Question 0.1 has been answered in the affirmative by Fujino-Mori [FM] for κ = 1. In this note we show that the answer is also affirmative for κ = 2. In the proof we do not 2000 Mathematics Subject Classification. 14E05, 14E25, 14Q20, 14J10.

Factorial threefold hypersurfaces

Abstract: Let $X$ be a hypersurface in $\mathbb{P}^{4}$ of degree $d$ that has at most isolated ordinary double points. We prove that $X$ is factorial in the case when $X$ has at most $(d-1)^{2}-1$ singular points.

Ekedahl-Oort strata contained in the supersingular locus and Deligne-Lusztig varieties

Abstract: We study the moduli space of principally polarized abelian varieties over fields of positive characteristic. In this paper we describe certain unions of Ekedahl-Oort strata contained in the supersingular locus in terms of Deligne-Lusztig varieties. As a corollary we show that each Ekedahl-Oort stratum contained in the supersingular locus is reducible except possibly for small p.

On Shimura curves in the Schottky locus

Abstract: We show that a given rational Shimura curve Y with strictly maximal Higgs field in the moduli space of g-dimensional abelian varieties does not generically intersect the Schottky locus for large g. We achieve this by using a result of Viehweg and Zuo which says that if Y parameterizes a family of curves of genus g, then the corresponding family of Jacobians is isogenous over Y to the g-fold product of a modular family of elliptic curves. After reducing the situation from the field of complex numbers to a finite field, we will see, combining the Weil and Sato-Tate conjectures, that this is impossible for large genus g.

Blow-analytic equivalence of two variable real analytic function germs

Abstract: Blow-analytic equivalence is a notion for real analytic function germs, in- troduced by Tzee-Char Kuo in order to develop real analytic equisingularity theory. In this paper we give complete characterisations of blow-analytic equivalence in the two dimensional case: in terms of the real tree model for the arrangement of real parts of Newton-Puiseux roots and their Puiseux pairs, and in terms of minimal resolutions. These characterisations show that in the two dimensional case the blow-analytic equiva- lence is a natural analogue of topological equivalence of complex analytic function germs. Moreover, we show that in the two-dimensional case the blow-analytic equivalence can be made cascade, and hence satisfies several geometric properties. It preserves, for instance, the contact orders of real analytic arcs. In the general n-dimensional case, we show that a singular real modification satisfies the arc-lifting property. A classical result of Burau (4) and Zariski (34) shows the embedded topological type of a plane curve singularity (X,0) ⊂ (C 2 ,0) is determined by the Puiseux pairs of each irreducible component and the intersection numbers of any pairs of distinct components. It can be shown, cf. (30), that the topological type of function germs f : (C 2 ,0) → (C,0) is completely characterised, also in the non-reduced case f = Q f di i , by the embedded topological type of its zero set and the multiplicities di of its irreducible components. In this paper we give a real analytic counterpart of these results and show that the two variable version of blow-analytic equivalence of Kuo is classified by invariants similar to Puiseux pairs, multiplicities of irreducible components, and intersection numbers. More- over we show several natural geometric properties of this equivalence, answering previously posed questions. In the main result of this paper we a give complete characterisation of blow-analytic equivalence classes of two variable real analytic function germs. Theorem 0.1. Let f : (R 2 ,0) → (R,0) and g : (R 2 ,0) → (R,0) be real analytic function germs. Then the following conditions are equivalent: (1) f and g are blow-analytically equivalent. (2) f and g have weakly isomorphic minimal resolution spaces. (3) The real tree models of f and g are isomorphic. Moreover if f and g are blow-analytically equivalent then they are equivalent by a cascade- blow-analytic homeomorphism.

General isotropic flags are general (for Grassmannian Schubert calculus)

Abstract: We show that general isotropic flags for odd-orthogonal and symplectic groups are general for Schubert calculus on the classical Grassmannian in that Schubert varieties defined by such flags meet transversally. This strengthens a result of Belkale and Kumar. Schubert varieties ­IEin a classical flag manifold G/P are given by a flag Eand a Schubert condition I (3). By Kleiman's Transversality Theorem (4), if the flags E 1

Some intersection numbers of divisors on toroidal compactifications of _{ℊ}

Abstract: We study the top intersection numbers of the boundary and Hodge class divisors on toroidal compactifications of the moduli space Ag of principally polarized abelian varieties and compute those numbers that live away from the stratum which lies over the closure of Ag 3 in the Satake compactification.

Hilbert’s theorem on positive ternary quartics: A refined analysis

Abstract: Let X be an integral plane quartic curve over a field k, let f be an equation for X. We first consider representations (∗) cf = p1p2 − p0 (where c ∈ k∗ and the pi are quadratic forms), up to a natural notion of equivalence. Using the general theory of determinantal varieties we show that equivalence classes of such representations correspond to nontrivial globally generated torsion-free rank one sheaves on X with a self-duality which are not exceptional, and that the exceptional sheaves are in bijection with the k-rational singular points of X. For k = C, the number of representations (∗) (up to equivalence) depends only on the singularities of X, and is determined explicitly in each case. In the second part we focus on the case where k = R and f is nonnegative. By a famous theorem of Hilbert, such f is a sum of three squares of quadratic forms. We use the Brauer group and Galois cohomology to relate identities (∗∗) f = p0 + p1 + p2 to (∗), and we determine the number of equivalence classes of representations (∗∗) for each f . Both in the complex and in the real definite case, our results are considerably more precise since they give the number of representations with any prescribed base locus.

Addendum to "Semistable reduction of G-bundles on curves''

Abstract: In this note we give a variation of the last step of the proof of the semistable reduction theorem of [J. Algebraic Geom. 17 (2008), 167-183], which allows us to remove the assumptions on the characteristic of the ground field.