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Showing papers in "Michigan Mathematical Journal in 2011"


Journal ArticleDOI
TL;DR: In this article, all linear Hodge classes on the moduli space of admissible covers with monodromy group G are associated to irreducible representations of G. In case G is cyclic and the representation is faithful, the evaluation is in terms of double Hurwitz numbers.
Abstract: Hodge classes on the moduli space of admissible covers with monodromy group G are associated to irreducible representations of G. We evaluate all linear Hodge integrals over moduli spaces of admissible covers with abelian monodromy in terms of multiplication in an associated wreath group algebra. In case G is cyclic and the representation is faithful, the evaluation is in terms of double Hurwitz numbers. In case G is trivial, the formula specializes to the well-known result of Ekedahl-Lando-Shapiro-Vainshtein for linear Hodge integrals over the moduli space of curves in terms of single Hurwitz numbers

67 citations


Journal ArticleDOI
TL;DR: Theorem 1. as mentioned in this paper shows that every simplicial automorphism of the free splitting graph of a free group Fn is induced by an outer automomorphism of Fn for n ≥ 3.
Abstract: We prove that every simplicial automorphism of the free splitting graph of a free group Fn is induced by an outer automorphism of Fn for n ≥ 3. In this note we consider the graph Gn of free splittings of the free group Fn of rank n ≥ 3. Loosely speaking, Gn is the graph whose vertices are non-trivial free splittings of Fn up to conjugacy, and where two vertices are adjacent if they are represented by free splittings admitting a common refinement. The group Out(Fn) of outer automorphisms of Fn acts simplicially on Gn. Denoting by Aut(Gn) the group of simplicial automorphisms of the free splitting graph, we prove: Theorem 1. The natural map Out(Fn)→ Aut(Gn) is an isomorphism for n ≥ 3. We briefly sketch the proof of Theorem 1. We identify Gn with the 1skeleton of the sphere complex Sn and observe that every automorphism of Gn extends uniquely to an automorphism of Sn. It is due to Hatcher [4] that the sphere complex contains an embedded copy of the spine Kn of Culler-Vogtmann space. We prove that the latter is invariant under Aut(Sn), and that the restriction homomorphism Aut(Sn) → Aut(Kn) is injective. The claim of Theorem 1 then follows from a result of Bridson-Vogtmann [1] which asserts that Out(Fn) is the full automorphism group of Kn. Before concluding this introduction we would like to point out that recently Martino and Francaviglia [9] have proved that Out(Fn) is also the full isometry group of Culler-Vogtmann space when the latter is endowed with the Lipschitz metric. While one could claim that Theorem 1 is the analogue of Ivanov’s theorem on the isometries of the curve complex [6], the result of Martino and Francaviglia is the analogue of Royden’s theorem on the isometries of Teichmuller space [10]. The first author was partially supported by M.E.C. grant MTM2006/14688 and NUI Galway’s Millennium Research Fund; the second author was partially supported by NSF grant DMS-0706878 and Alfred P. Sloan Foundation.

37 citations


Journal ArticleDOI
TL;DR: In this paper, the authors construct many more examples of positive entropy automorphisms on rational complex projective plane P2 by using the theory of infinite Coxeter groups, some results of Nagata [N1] about Cremona transformations, and important properties of plane cubic curves.
Abstract: Every automorphism of the complex projective plane P2 is linear and therefore behaves quite simply when iterated. It is natural to seek other rational complex surfaces—for instance, those obtained from P2 by successive blowing up—that admit automorphisms with more interesting dynamics. Until recently, very few examples with positive entropy seem to have been known (see e.g. the introduction to [Ca]). Bedford and Kim [BeK2] found some new examples by studying an explicit family of Cremona transformations—namely, birational self-maps of P2. McMullen [Mc] gave a more synthetic construction of some similar examples. To this end he used the theory of infinite Coxeter groups, some results of Nagata [N1; N2] about Cremona transformations, and important properties of plane cubic curves. In this paper, we construct many more examples of positive entropy automorphisms on rational surfaces. Whereas [Mc] seeks automorphisms with essentially arbitrary topological behavior, we limit our search to automorphisms that might conceivably be induced by Cremona transformations of polynomial degree 2 (quadratic transformations for short). This restriction allows us be more explicit about the automorphisms we find and to make do with less technology, using only the group law for cubic curves (suitably interpreted when the curve is singular or reducible) in place of Coxeter theory and Nagata’s theorems. A quadratic transformation f : P2 → P2 always acts by blowing up three (indeterminacy) points I(f ) = {p+ 1 ,p2 ,p3 } in P2 and blowing down the (exceptional) lines joining them. Typically, the points and the lines are distinct, but in general they can occur with multiplicity (see Section 1.2). Regardless, f −1 is also a quadratic transformation and I(f −1) = {p− 1 ,p2 ,p3 } consists of the images of the three exceptional lines. Under certain fairly checkable circumstances, a quadratic transformation f will lift to an automorphism of some rational surface X obtained from P2 by a finite sequence of point blowups. Namely, suppose there are integers n1, n2, n3 ∈N and a permutation σ ∈ 3 such that f nj−1(p− j ) = p+ σj for j = 1, 2, 3. We assume that

33 citations


Journal ArticleDOI
TL;DR: In this paper, a new type of toric fiber product theorem was introduced to prove the full conjecture by introducing a new Toric Fiber Product Theorem (TFP) and showed that the cut ideal is generated by quadrics if and only if the graph is free of K4minors.
Abstract: Cut ideals are used in algebraic statistics to study statistical models defined by graphs. Intuitively, topological restrictions on the graphs should imply structural statements about the corresponding cut ideals. Several theorems and many computer calculations support that. Sturmfels and Sullivant conjectured that the cut ideal is generated by quadrics if and only if the graph is free of K4-minors. Parts of the conjecture has been resolved by Brennan and Chen, and later by Nagel and Petrovic. We prove the full conjecture by introducing a new type of toric fiber product theorem.

30 citations


Journal ArticleDOI
TL;DR: In this paper, the authors generalized the work of Favre to higher dimensions and showed that a monomial selfmap can be made stable by refining the underlying fan of a toric variety.
Abstract: A monomial (i.e. equivariant) selfmap of a toric variety is called stable if its action on the Picard group commutes with iteration. Generalizing work of Favre to higher dimensions, we show that under suitable conditions, a monomial map can be made stable by refining the underlying fan. In general, the resulting toric variety has quotient singularities; in dimension two we give criteria for when it can be chosen smooth, as well as examples when it cannot.

24 citations


Journal ArticleDOI
TL;DR: In this article, the ex-istence of all rank 2 ACM vector bundles on a smooth non-hyperelliptic prime Fano threefold X was proved by deformation of semistable sheaves.
Abstract: Given a smooth non-hyperelliptic prime Fano threefold X, we prove the ex- istence of all rank 2 ACM vector bundles on X by deformation of semistable sheaves. We show that these bundles move in generically smooth components of the corresponding moduli space. We give applications to pfaffian representations of quartic t in P 4 and cubic hypersurfaces of a smooth quadric of P 5 .

24 citations


Journal ArticleDOI
TL;DR: In this paper, the existence of a Chow-Kuenneth decomposition for a rational homogeneous bundle over a smooth variety is investigated and the same conclusion holds for a class of log homogeneous varieties, studied by Brion.
Abstract: In this paper, we investigate Murre's conjecture on the existence of a Chow--Kuenneth decomposition for a rational homogeneous bundle $Z\\to S$ over a smooth variety, defined over complex numbers. Chow-K\\\"unneth decomposition is exhibited for $Z$ whenever $S$ has a Chow--Kuenneth decomposition. The same conclusion holds for a class of log homogeneous varieties, studied by M. Brion.

23 citations


Journal ArticleDOI
TL;DR: In this article, the Veronese map is used to classify real-analytic CR maps between any hyperquadric in Ω(C^2$ and any hyper quadric in √ C^3$ resulting in a finite list of equivalence classes, and all degree-two CR maps of spheres are spherically equivalent to a monomial map.
Abstract: We prove and organize some results on the normal forms of Hermitian operators composed with the Veronese map. We apply this general framework to prove two specific theorems in CR geometry. First, extending a theorem of Faran, we classify all real-analytic CR maps between any hyperquadric in $\C^2$ and any hyperquadric in $\C^3$, resulting in a finite list of equivalence classes. Second, we prove that all degree-two CR maps of spheres in all dimensions are spherically equivalent to a monomial map, thus obtaining an elegant classification of all degree-two CR sphere maps.

23 citations



Journal ArticleDOI
TL;DR: In this paper, the weak factorization theorem of Wlodarczyk and tools developed by Friedlander, Lawson, Lima-Filho and others are used to define new birational invariants for a projective manifold.
Abstract: New birational invariants for a projective manifold are defined by using Lawson homology. These invariants can be highly nontrivial even for projective threefolds. Our techniques involve the weak factorization theorem of Wlodarczyk and tools developed by Friedlander, Lawson, Lima-Filho and others. A blowup formula for Lawson homology is given in a separate section. As an application, we show that for each n\geq 5, there is a smooth rational variety X of dimension n such that the Griffiths groups Griff}_p(X) are infinitely generated even modulo torsion for all p with 2\leq p\leq n-3.

20 citations


Journal ArticleDOI
TL;DR: For real numbers x ≥ 1 and A ≥ 1, put G(x,A) := #{n ≤ x : gcd(n, σ(n)) > A}.
Abstract: A natural number n is called perfect if σ(n) = 2n and multiply perfect whenever σ(n) is a multiple of n. In 1956, Erdős published improved upper bounds on the counting functions of the perfect and multiply perfect numbers [2]. These estimates were soon superseded by a theorem of Wirsing [15] (Theorem B below), but Erdős’s methods remain of interest as they are applicable to more general questions concerning the distribution of gcd(n, σ(n)). Erdős describes some applications of this type (op. cit.) but omits the proofs. In this paper we prove corrected versions of his results, and we establish some new results in the same direction. For real numbers x ≥ 1 and A ≥ 1, put G(x,A) := #{n ≤ x : gcd(n, σ(n)) > A}.

Journal ArticleDOI
TL;DR: In this article, the Weil Petersson geometry of a family of Riemann surfaces with conical hyperbolic metrics was studied, and the authors studied the WeIL Petersson geometrical properties of a set of conical surfaces with a conical geometry.
Abstract: We study the Weil Petersson geometry of a family of Riemann surfaces with conical hyperbolic metrics

Journal ArticleDOI
TL;DR: In this paper, a closed formula for recovering a non-hyperelliptic genus three curve from its period matrix was given, and some identities between Jacobian Nullwerte in dimension three were derived.
Abstract: We give a closed formula for recovering a non-hyperelliptic genus three curve from its period matrix, and derive some identities between Jacobian Nullwerte in dimension three.

Journal ArticleDOI
TL;DR: In this article, the authors classify smooth complex projective varieties with dimension 2s+1 and a linear subspace of dimension 2 s+1 whose normal bundle is numerically effective.
Abstract: We classify smooth complex projective varieties $X \subset \proj^N$ of dimension $2s+1$ containing a linear subspace $\Lambda$ of dimension $s$ whose normal bundle $N_{\Lambda/X}$ is numerically effective.


Journal ArticleDOI
TL;DR: In this paper, the authors studied the Fano surfaces of lines of cubic three-folds that contain 12 or 30 elliptic curves and determined their Picard number and computed a basis of the N\'eron-Severi group of the Fermat cubic threefold.
Abstract: Curves of low genus on a surface carry important informations on that surface. We study the Fano surfaces of lines of cubic threefolds that contain 12 or 30 elliptic curves. We determine their Picard number and compute a basis of the N\'eron-Severi group of the Fano surface of the Fermat cubic threefold.


Journal ArticleDOI
TL;DR: In this paper, the authors consider a smooth projective curve of genus g and a line bundle on the curve, where the morphism p ∗(p ∗ 1L ⊗ p2L) = L ⊕ L−, where L± are invariant and anti-invariant line bundles with respect to the involution (x, y) 7→ (y, x).
Abstract: Let X be a smooth, projective curve of genus g and let L be a line bundle on X. Consider the product X ×X, with the projections p1, p2 to the factors, and the natural morphism p to the symmetric product X(2). One has p∗(p ∗ 1L ⊗ p2L) = L ⊕ L−, where L± are the invariant and anti-invariant line bundles with respect to the involution (x, y) 7→ (y, x). One has H(L) ∼= SymH0(L) and H0(L−) ∼= ∧H(L). Restriction to the diagonal of X(2) gives rise to two maps

Journal ArticleDOI
TL;DR: In this article, the difference between the Hodge polynomials of the singular and resp. generic member of a pencil of hypersurfaces in a projective manifold was expressed by using a stratification of singular member described in terms of the data of the pencil.
Abstract: We express the difference between the Hodge polynomials of the singular and resp. generic member of a pencil of hypersurfaces in a projective manifold by using a stratification of the singular member described in terms of the data of the pencil. In particular, if we assume trivial monodromy along all strata in the singular member of the pencil, our formula computes this difference as a weighted sum of Hodge polynomials of singular strata, the weights depending only on the Hodge-type information in the normal direction to the strata. This extends previous results (cf. [19]) which related the Euler characteristics of the generic and singular members only of generic pencils, and yields explicit formulas for the Hodge χy-polynomials of singular hypersurfaces.



Journal ArticleDOI
TL;DR: In this article, the fundamental groups of maximizing plane sextics with a type E6 singular point were derived and a classification up to equisingular deformation was given up to the point of equisingularity.
Abstract: We give a classification up to equisingular deformation and compute the fundamental groups of maximizing plane sextics with a type E6 singular point.

Journal ArticleDOI
TL;DR: In this paper, the authors extend two results on Chow (semi-)stability to positive characteristics, namely, the stability of non-singular projective hypersurfaces of degree at least three and the criterion by Y. Lee in terms of log canonical thresholds.
Abstract: We extend two results on Chow (semi-)stability to positive characteristics. One is on the stability of non-singular projective hypersurfaces of degree at least three, and the other is the criterion by Y. Lee in terms of log canonical thresholds. Some properties of log-canonicity in positive characteristics are discussed with a couple of examples, in connection with the proof of the latter one. It is also proven in Appendix that the sum of Chow (semi- )stable cycles are again Chow (semi-)stable.


Journal ArticleDOI
TL;DR: In this paper, the authors studied the restricted volume along subvarieties of line bundles with non-negative Kodaira-Iitaka dimension and proved that the former is non-zero if and only if the latter is big.
Abstract: We study the restricted volume along subvarieties of line bundles with non-negative Kodaira-Iitaka dimension. Our main interest is to compare it with a similar notion defined in terms of the asymptotic multiplier ideal sheaf, with which it coincides in the big case. We shall prove that the former is non-zero if and only if the latter is. We then study inequalities between them and prove that if they coincide on every very general curve the line bundle must have zero Kodaira-Iitaka dimension or be big. Let X be a smooth projective variety, L a divisor or a line bundle on X with nonnegative Kodaira-Iitaka dimension: κ(L) ≥ 0. Let V ⊂ X be a subvariety of dimV = d > 0 such that V 6⊂ SBs (L), where SBs (L) := ⋂ m>0 Bs |mL| is the stable base locus. We denote by H(X|V,mL) = Image [H(X,mL) −→ H(V,mL)] the image of restriction maps. The restricted volume of L along V is defined to be

Journal ArticleDOI
TL;DR: In this article, the authors generalize the notion of isolated singularities to non-isolated hypersurface singularities and show that Whitney equisingularity is equivalent to constancy of a certain selection of invariants from two distinct generalizations of the µ-sequence.
Abstract: In the study of equisingularity of isolated singularities we have the classical theorem of Briancon, Speder and Teissier which states that a family of iso- lated hypersurface singularities is Whitney equisingular if and only if the µ - sequence for a hypersurface is constant in the family. In this paper we general- ize to non-isolated hypersurface singularities. By assuming non-contractibility of strata of a Whitney stratification of the non-isolated singularities outside the origin we show that Whitney equisingularity of a family is equivalent to constancy of a certain selection of invariants from two distinct generalizations of the µ -sequence. Applications of this theorem to equisingularity of more general mappings are given. AMS Mathematics Subject Classification 2000 : 32S15, 32S30, 32S60.

Journal ArticleDOI
TL;DR: In this article, the generalized Oka-Grauert principle for 1-convex manifolds has been shown to be applicable to complex manifold transformations on a neighborhood of the exceptional set.
Abstract: This paper presents a proof of the generalized Oka-Grauert principle for 1-convex manifolds: Every continuous mapping from a 1-convex manifold X to a complex manifold Y which is already holomorphic on a neighborhood of the exceptional set is homotopic to a holomorphic one provided that either Y satisfies CAP or we are free to change the complex structure on X.


Journal ArticleDOI
TL;DR: In this paper, the authors obtained an infinite family of simply connected mutually diffeomorphic 4-manifolds coming from a pair of inequivalent Kanenobu knots, each of which admits more than one inequivalent Lefschetz fibration structures of the same generic fiber.
Abstract: In this article we study Lefschetz fibration structures on knot surgery 4-manifolds obtained from an elliptic surface E(2) using Kanenobu knots $K$. As a result, we get an infinite family of simply connected mutually diffeomorphic 4-manifolds coming from a pair of inequivalent Kanenobu knots. We also obtain an infinite family of simply connected symplectic 4-manifolds, each of which admits more than one inequivalent Lefschetz fibration structures of the same generic fiber.

Journal ArticleDOI
TL;DR: In this paper, a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space was proved for 1 n = 7, where n is the number of nodes in the collared domain.
Abstract: In this paper we prove a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space: for 1 n = 7.