Ulrich bundles on ruled surfaces

In this short note, we study the existence problem for Ulrich bundles on polarized ruled surfaces, focusing our attention on the smallest possible rank. We show that existence of Ulrich line bundles occurs if and only if the coefficient α of the minimal section in the numerical class of the polarization equals one. For other polarizations, we prove the existence of rank two Ulrich bundles.     

Ulrich bundles on rational surfaces with an anticanonical pencil

Ulrich bundles are the simplest sheaves from the viewpoint of cohomology tables. Eisenbud and Schreyer conjectured that every projective variety carries an Ulrich bundle, which implies it has the same cone of cohomology tables as the projective space of same dimension. In this paper we show the existence of stable rank 2 Ulrich bundles on rational surfaces with an anticanonical pencil, under a mild Brill–Noether assumption by using Lazarsfeld–Mukai bundles. As a consequence, the Chow form of a surface admitting a rank 2 Ulrich bundle of the form we construct is known to be the Pfaffian of a skew-symmetric matrix.     

Ulrich and aCM bundles from invariant theory

We use certain special prehomogeneous representations of algebraic groups in order to construct aCM vector bundles, possibly Ulrich, on certain families of hypersurfaces. Among other results, we show that a general cubic hypersurface of dimension seven admits an indecomposable Ulrich bundle of rank nine, and that a general quartic fourfold admits an unsplit aCM bundle of rank six.     

Ulrich bundles on quartic surfaces with Picard number 1

In this note, we prove that there exist stable Ulrich bundles of every even rank on a smooth quartic surface $X?{\mathbb{P}}^3$ with Picard number 1.     

Stable Ulrich bundles on Fano threefolds with Picard number 2

In this paper, we consider the existence problem of rank one and two stable Ulrich bundles on imprimitive Fano 3-folds obtained by blowing-up one of ${\mathbb{P}}^3$, Q (smooth quadric in ${\mathbb{P}}^4$), $V_3$ (smooth cubic in ${\mathbb{P}}^4$) or $V_4$ (complete intersection of two quadrics in ${\mathbb{P}}^5$) along a smooth irreducible curve. We prove that the only class which admits Ulrich line bundles is the one obtained by blowing up a genus 3, degree 6 curve in ${\mathbb{P}}^3$. Also, we prove that there exist stable rank two Ulrich bundles with $c_1=3H$ on a generic member of this deformation class.     

Ulrich line bundles on Enriques surfaces with a polarization of degree four

In this paper, we prove the existence of an Enriques surface with a polarization of degree four with an Ulrich bundle of rank one. As a consequence, we prove that general polarized Enriques surfaces of degree four, with the same numerical polarization class, carry Ulrich line bundles.     

Multiplicities and Hilbert functions under blowing up

In this note we study the behaviour of Hilbert-functions and multiplicities of a local ring R under blowing up an ideal A of R such that R/A need not be regular or even a domain, using the result of Singh ([10]) about Hilbert functions under quadratic transformations. If A is a regular equimultiple prime ideal (and R quasi-unmixed), it is shown that the multiplicity cannot increase by blowing up A. This is an unpublished result by E. Dade ([2]), and actually our proof is in part inspired by his methods. The other main ingredient is the use of a generalized Hilbert-function introduced in [5].I would like to thank S.S. Abhyankar, M. Herrmann and J. Lipman for helpful discussions.     

The Hermite-Einstein equation and stable principal bundles (an updated survey)

This is a survey of work done in the past two decades relating a basic equation of general relativity and quantum field theory with the theory of stable vector bundles and stable principal bundles, as well as providing moduli spaces for such objects, and compactifying them. This is an updated version of a previous one which met an internal publication with occasion of the memorial homage in October 2003 to prof. José María Fraile, a friend and analyst member of our faculty then departed. Since perhaps this short exposition of facts may help to outsiders interested in the subject, or help people just entering the field, we have found it convenient to take advantage of the lecture of the second author on this subject in Ouro Preto 2007, to provide in this volume of wider diffusion and addressed to geometers an updated version of that exposé.     

The Lefschetz property, formality and blowing up in symplectic geometry

In this paper we study the behaviour of the Lefschetz property under the blow-up construction. We show that it is possible to reduce the dimension of the kernel of the Lefschetz map if we blow up along a suitable submanifold satisfying the Lefschetz property. We use this, together with results about Massey products, to construct compact nonformal symplectic manifolds satisfying the Lefschetz property.

     

Semistable and numerically effective principal (Higgs) bundles

We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class.In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian–Yang–Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.     

Cheeger-gromoll type metrics on the (1,1)-tensor bundles

Using a Riemannian metric on a differentiable manifold, a Cheeger-Gromoll type metric is introduced on the (1,1)-tensor bundle of the manifold. Then the Levi-Civita connection, Riemannian curvature tensor, Ricci tensor, scalar curvature and sectional curvature of this metric are calculated. Also, a para-Nordenian structure on the the (1,1)-tensor bundle with this metric is constructed and the geometric properties of this structure are studied.     

Well-posedness of a Dissolution-growth problem on an unbounded domain erratum

We consider a spherical grain which may be growing by accretion or dissolving in a dilute solution of the same substance where one also has reaction and diffusion; the resulting free boundary problem was shown to be well-posed. The paper as originally printed was inadvertently truncated and this Erratum consists of Section 4 (on additional regularity) and the Bibliography.     

On the blowing up of solutions to quantum hydrodynamic models on bounded domains

The blow-up in finite time for the solutions to the initial-boundary value problem associated to the multi-dimensional quantum hydrodynamic model in a bounded domain is proved. The model consists on conservation of mass equation and a momentum balance equation equivalent to a compressible Euler equations corrected by a dispersion term of the third order in the momentum balance. The proof is based on a priori estimates for the energy functional for a new observable constructed with an auxiliary function, and it is shown that, under suitable boundary conditions and assumptions on the initial data, the solution blows up after a finite time. I.M. Gamba is supported by NSF-DMS0507038. M.P. Gualdani acknowledges partial support from the Deutsche Forschungsgemeinschaft, grants JU359/5 and was partially supported under the Feodor Lynen Research fellowship. P. Zhang is partially supported by the NSF of China under Grant 10525101 and 10421101, and the innovation grant from the Chinese Academy of Sciences. Part of the work was done when P. Zhang visited the Department of Mathematics of Texas University at Austin, the author would like to thank the hospitality of the department. Support from the Institute for Computational Engineering and Sciences at the University of Texas at Austin is also gratefully acknowledged.     

On the blowing up of solutions to quantum hydrodynamic models on bounded domains

The blow-up in finite time for the solutions to the initial-boundary value problem associated to the multi-dimensional quantum hydrodynamic model in a bounded domain is proved. The model consists on conservation of mass equation and a momentum balance equation equivalent to a compressible Euler equations corrected by a dispersion term of the third order in the momentum balance. The proof is based on a priori estimates for the energy functional for a new observable constructed with an auxiliary function, and it is shown that, under suitable boundary conditions and assumptions on the initial data, the solution blows up after a finite time.