On the Hilbert scheme of curves in higher-dimensional projective space

In this paper we prove that, for any $n\ge 3$, there exist infinitely many $r\in \N$ and for each of them a smooth, connected curve $C_r$ in $\P^r$ such that $C_r$ lies on exactly $n$ irreducible components of the Hilbert scheme \hilb(\P^r). This is proven by reducing the problem to an analogous statement for the moduli of surfaces of general type.Comment: latex, 12 pages, no figure

Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane Curves

We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives

On compatibility between isogenies and polarisations of abelian varieties

We discuss the notion of polarised isogenies of abelian varieties, that is, isogenies which are compatible with given principal polarisations. This is motivated by problems of unlikely intersections in Shimura varieties. Our aim is to show that certain questions about polarised isogenies can be reduced to questions about unpolarised isogenies or vice versa. Our main theorem concerns abelian varieties B which are isogenous to a fixed abelian variety A. It establishes the existence of a polarised isogeny A to B whose degree is polynomially bounded in n, if there exist both an unpolarised isogeny A to B of degree n and a polarised isogeny A to B of unknown degree. As a further result, we prove that given any two principally polarised abelian varieties related by an unpolarised isogeny, there exists a polarised isogeny between their fourth powers. The proofs of both theorems involve calculations in the endomorphism algebras of the abelian varieties, using the Albert classification of these endomorphism algebras and the classification of Hermitian forms over division algebras

Alternating groups and moduli space lifting Invariants

Main Theorem: Spaces of r-branch point 3-cycle covers, degree n or Galois of degree n!/2 have one (resp. two) component(s) if r=n-1 (resp. r\ge n). Improves Fried-Serre on deciding when sphere covers with odd-order branching lift to unramified Spin covers. We produce Hurwitz-Torelli automorphic functions on Hurwitz spaces, and draw Inverse Galois conclusions. Example: Absolute spaces of 3-cycle covers with +1 (resp. -1) lift invariant carry canonical even (resp. odd) theta functions when r is even (resp. odd). For inner spaces the result is independent of r. Another use appears in, http://www.math.uci.edu/~mfried/paplist-mt/twoorbit.html, "Connectedness of families of sphere covers of A_n-Type." This shows the M(odular) T(ower)s for the prime p=2 lying over Hurwitz spaces first studied by, http://www.math.uci.edu/~mfried/othlist-cov/hurwitzLiu-Oss.pdf, Liu and Osserman have 2-cusps. That is sufficient to establish the Main Conjecture: (*) High tower levels are general-type varieties and have no rational points.For infinitely many of those MTs, the tree of cusps contains a subtree -- a spire -- isomorphic to the tree of cusps on a modular curve tower. This makes plausible a version of Serre's O(pen) I(mage) T(heorem) on such MTs. Establishing these modular curve-like properties opens, to MTs, modular curve-like thinking where modular curves have never gone before. A fuller html description of this paper is at http://www.math.uci.edu/~mfried/paplist-cov/hf-can0611591.html .Comment: To appear in the Israel Journal as of 1/5/09; v4 is corrected from proof sheets, but does include some proof simplification in \S

Up and Down and Back Again: Troubled Childhood Childhood Notwithstanding, Washington\u27s Stand Alone Estate Tax Deserves to be Defended

This Comment evaluates the history of Washington\u27s estate tax from the pre-2005 frozen scheme, through the Supreme Court\u27s analysis and mandate in Estate of Hemphill v. State, and up to the legislation enacted in May 2005. Part II provides a background on EGTRRA and evaluates the extent of its changes nationwide. Part III critically reviews Washington\u27s estate tax history, and examines both the seminal Initiative 402 and the legislative history supporting the shift away from federal conformation. Part IV analyzes how the court\u27s 2005 ruling provided the catalyst for legislative change, and provides a summary of Hemphill and the arguments presented therein. Part V argues that Senate Bill 6096 is a sound step towards dealing with the inevitable fiscal issues resulting both from Washington\u27s pre-2005 scheme and from the choice Washington had to make in light of Hemphill. Part VI evaluates the ramifications and problems that the change to EGTRRA would have inflicted and that Senate Bill 6096 specifically avoids, and encourages the legislature to treat the bill as but one step of an ongoing process of proactive taxation. Part VII concludes the Comment with the point that the newly enacted stand-alone tax is the best way to square the interests of all parties involved and notes that, although it may not be an appropriate permanent fix, Senate Bill 6096 is a positive step for our state and the legislature should be encouraged to continue improving upon it

Geometric collections and Castelnuovo-Mumford Regularity

The paper begins by overviewing the basic facts on geometric exceptional collections. Then, we derive, for any coherent sheaf \cF on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to \cF and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties $X$ with a geometric collection $\sigma$. We define the notion of regularity of a coherent sheaf \cF on $X$ with respect to $\sigma$. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we show that in case of coherent sheaves on \PP^n and for a suitable geometric collection of coherent sheaves on \PP^n both notions of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a smooth quadric hypersurface Q_n \subset \PP^{n+1} ($n$ odd) with respect to a suitable geometric collection and we compare it with the Castelnuovo-Mumford regularity of their extension by zero in \PP^{n+1}.Comment: To appear in Math. Proc. Cambridg

Spectrum of the quantum Neumann model

We study numerically the spectrum and eigenfunctions of the quantum Neumann model, illustrating some general properties of a non trivial integrable model.Comment: 12 pages, 5 figures. Expanded introduction and references to put our work in the proper historical contex

Developing transferable management skills through Action Learning

There has been increasing criticism of the relevance of the Master of Business Administration (MBA) in developing skills and competencies. Action learning, devised to address problem-solving in the workplace, offers a potential response to such criticism. This paper offers an insight into one university’s attempt to integrate action learning into the curriculum. Sixty-five part-time students were questioned at two points in their final year about their action learning experience and the enhancement of relevant skills and competencies. Results showed a mixed picture. Strong confirmation of the importance of selected skills and competencies contrasted with weaker agreement about the extent to which these were developed by action learning. There was, nonetheless, a firm belief in the positive impact on the learning process. The paper concludes that action learning is not a panacea but has an important role in a repertoire of educational approaches to develop relevant skills and competencies

Semiclassical Strings in AdS_5 x S^5 and Automorphic Functions

Using AdS/CFT we derive from the folded spinning string ordinary differential equations for the anomalous dimension of the dual N=4 SYM twist-two operators at strong coupling. We show that for large spin the asymptotic solutions have the Gribov-Lipatov recirocity property. To obtain this result we use a hidden modular invariance of the energy-spin relation of the folded spinning string. Further we identify the Moch-Vermaseren-Vogt (MVV) relations, which were first recognized in plain QCD calculations, as the recurrence relations of the asymptotic series ansatz.Comment: 4 page

Ultradiscretization of the solution of periodic Toda equation

A periodic box-ball system (pBBS) is obtained by ultradiscretizing the periodic discrete Toda equation (pd Toda eq.). We show the relation between a Young diagram of the pBBS and a spectral curve of the pd Toda eq.. The formula for the fundamental cycle of the pBBS is obtained as a colloraly.Comment: 41 pages; 7 figure