Generalisation of the Bogomolov-Miyaoka-Yau Inequality to Singular Surfaces

The paper considers pairs (X, B) where X is a normal projectivesurface over C, and B is a Q-divisor whose coefficients are1 or 1–1/m for some natural number m. A log canonicalsingularity on such a pair is a quotient by a finite or infinitegroup, so if (X, B) has log canonical singularities, the orbifoldEuler number e_orb(X, B) can be defined. The main result is aBogomolov-Miyaoka-Yau-type inequality which implies that if(X, B) has log canonical singularities and (X, K_X + B) 0 then(K_X+B)² 3e_orb(X, B). The actual inequality proved is somewhatstronger and it also implies all the previously published versionsof the Bogomolov-Miyaoka-Yau inequality. The proof involvesthe Log Minimal Model Program, Q-sheaves when K_X+B is nef, anda study of the changes in the two sides of the inequality undera contraction. The paper also contains a further generalisationwhere the coefficients of B can be arbitrary rational numbersin [0, 1], a different condition is imposed on the singularitiesand K_X+B is required to be nef. Some applications of the inequalitiesare also given, for example, estimating the number of singularitiesor certain kinds of configurations of curves on surfaces. 1991Mathematics Subject Classification: 14J17, 14J60, 14C17.     

The Global Mckay-Ruan Correspondence Via Motivic Integration

The purpose of this paper is to show how the methods of motivicintegration of Kontsevich, Denef–Loeser (Invent. Math.135 (1999) 201–232 and Compositio Math. 131 (2002) 267–290)and Looijenga (Astérisque 276 (2002) 267–297) canbe adapted to prove the McKay–Ruan correspondence, a generalizationof the McKay–Reid correspondence to orbifolds that arenot necessarily global quotients. 2000 Mathematics Subject Classification14A20, 14E15, 14F43.     

Linear Systems and Quotients of Projective Space

A complete characterization of the categorical quotients of(P¹)ⁿ by the diagonal action of SL(2, C) with respect to anypolarization is given by M. Polito, in ‘SL(2, C)-quotientsde (P¹)ⁿ’, C. R. Acad. Sci. Paris Sér. I 321 (1995)1577–1582. In this paper, these categorical quotientsare obtained by certain linear systems on P^n–3, dependingon the given polarization. 2000 Mathematics Subject Classification14L24, 14L30     

Stability of the Picard Bundle

Let X be a non-singular algebraic curve of genus g 2, n 2an integer, a line bundle over X of degree d > 2n(g –1) with (n,d) = 1 and M the moduli space of stable bundles ofrank n and determinant over X. It is proved that the Picardbundle W is stable with respect to the unique polarisation ofM. 2000 Mathematics Subject Classification 14H60, 14J60.     

On the Poles of Maximal Order of the Topological Zeta Function

The global and local topological zeta functions are singularityinvariants associated to a polynomial f and its germ at 0, respectively.By definition, these zeta functions are rational functions inone variable, and their poles are negative rational numbers.In this paper we study their poles of maximal possible order.When f is non-degenerate with respect to its Newton polyhedron,we prove that its local topological zeta function has at mostone such pole, in which case it is also the largest pole; wegive a similar result concerning the global zeta function. Moreover,for any f we show that poles of maximal possible order are alwaysof the form –1/N with N a positive integer. 1991 MathematicsSubject Classification 14B05, 14E15, 32S50.     

Complete Sets of Relations in the Cohomology Rings of Moduli Spaces of Holomorphic Bundles and Parabolic Bundles Over a Riemann Surface

The cohomology ring of the moduli space of stable holomorphicvector bundles of rank n and degree d over a Riemann surfaceof genus g > 1 has a standard set of generators when n andd are coprime. When n = 2 the relations between these generatorsare well understood, and in particular a conjecture of Mumford,that a certain set of relations is a complete set, is knownto be true. In this article generalisations are given of Mumford'srelations to the cases when n > 2 and also when the bundlesare parabolic bundles, and these are shown to form completesets of relations. 2000 Mathematics Subject Classification 14H60.     

Conditions for the Solvability of Systems of Two and Three Additive Forms Over p-Adic Fields

This paper is concerned with non-trivial solvability in p-adicintegers of systems of two and three additive forms. Assumingthat the congruence equation ax^k + by^k + cz^k d (modp) has asolution with xyz 0(modp) we have proved that any system oftwo additive forms of odd degree k with at least 6k + 1 variables,and any system of three additive forms of odd degree k withat least 14k + 1 variables always has non-trivial p-adic solutions,provided p does not divide k. The assumption of the solubilityof the congruence equation above is guaranteed for example ifp > k⁴. In the particular case of degree k = 5 we have proved the followingresults. Any system of two additive forms with at least n variablesalways has non-trivial p-adic solutions provided n 31 and p> 101 or n 36 and p > 11. Furthermore any system of threeadditive forms with at least n variables always has non-trivialp-adic solutions provided n 61 and p > 101 or n 71 andp > 11. 2000 Mathematics Subject Classification 11D72, 11D79.     

Exact Sequences of Semistable Vector Bundles on Algebraic Curves

Let X be a smooth complex projective curve of genus g 1. Ifg 2, then assume further that X is either bielliptic or withgeneral moduli. Fix integers r, s, a, b with r > 1, s >1 and as br. Here we prove the existence of an exact sequence [formula] of semistable vector bundles on X with rk(H) = r, rk(Q) = s,deg(H) = a and deg(Q) = b. 1991 Mathematics Subject Classification14H60.     

Topologie Des (M - 2)-Courbes Reelles Symetriques

Let X be a non-singular real algebraic curve in CP² of evendegree. In this paper a refinement is proved of a theorem ofKharlamov about (M – 2)-curves that are invariants underthe projective involution. In particular, if the (M –2)-symmetric curve X satisfies the Arnold congruence, then eitherX or its twin is a separating curve. 2000 Mathematics SubjectClassification 14P25.     

Deformation of Complete Intersections in the Plane

We study deformations of zero-dimensional complete intersectionsin the plane, and prove the following results. (1) Two complexnon-singular curves intersecting at r points with multiplicitiesd₁,...,d_r can be deformed into curves intersecting (at somepoints) with multiplicities d'₁,...,d'_s which are arbitraryprescribed partitions of the numbers d₁,...,d_r. (2) Two realcurves intersecting with multiplicity at most 2 at each of theirreal common points can be deformed so that all real multipleintersection points split into real simple intersection points.1991 Mathematics Subject Classification 14M10, 14P05.     

Cm Elliptic Curves and Bicolorings of Steiner Triple Systems

It is shown that a necessary condition for the existence ofa bicolored Steiner triple system of order n is that n can bewritten in the form A²+3B² for integers A and B. In the casewhen n=q is either a prime congruent to 1 mod 3, or the squareof a prime congruent to 2 mod 3, it is shown that the numbersof colored vertices in the triple system would be unique, andare given by the number of points on specific twists of theCM elliptic curve y²=x³–1 over the finite field F_q. 2000Mathematics Subject Classification 05B07, 11G20, 14G15 (primary);11G15, 14K22 (secondary).     

Integral Estimates for Transport Densities

The integration-by-parts methods introduced in this paper improveupon the L^p estimates on transport densities given in the recentpaper by L. De Pascale and A. Pratelli (Calc. Var. Partial DifferentialEquations 14 (2002) 249–274). 2000 Mathematics SubjectClassification 35Q99, 35B99.     

Zeta Functions for Curves and Log Canonical Models

The topological zeta function and Igusa's local zeta functionare respectively a geometrical invariant associated to a complexpolynomial f and an arithmetical invariant associated to a polynomialf over a p-adic field. When f is a polynomial in two variables we prove a formula forboth zeta functions in terms of the so-called log canonicalmodel of f^-1{0} in A². This result yields moreover a conceptualexplanation for a known cancellation property of candidate polesfor these zeta functions. Also in the formula for Igusa's localzeta function appears a remarkable non-symmetric ‘q-deformation’of the intersection matrix of the minimal resolution of a Hirzebruch-Jungsingularity. 1991 Mathematics Subject Classification: 32S5011S80 14E30 (14G20)     

Some Sharp Bounds for the Cone Multiplier of Negative Order in R3

This paper considers the cone multiplier operator which is definedby where and . For –3/2 < µ < –3/14, sharp L^p – L^q estimatesand endpoint estimates for S^µ are obtained. 2000 MathematicsSubject Classification 42B15 (primary).     

Asymptotics for the Heat Content of a Planar Region with a Fractal Polygonal Boundary

Let k 3 be an integer. For 0<s<1, let D_s R² be the setthat is constructed iteratively as follows. Take a regular openk-gon with sides of unit length, attach regular open k-gonswith sides of length s to the middles of the edges, and so on.At each stage of the iteration the k-gons that are added area factor s smaller than the previous generation and are attachedto the outer edges of the family grown so far. The set D_s isdefined to be the interior of the closure of the union of allthe k-gons. It is easy to see that there must exist some s_k> 0 such that no k-gons overlap if and only if 0 < s s_k. We derive an explicit formula for s_k. The set D_s is open, bounded, connected and has a fractal polygonalboundary. Let denote the heat content of D_s at time t when D_s initially has temperature 0and D_s is kept at temperature 1. We derive the complete short-timeexpansion of up to terms that are exponentially small in 1/t. It turns out that there arethree regimes, corresponding to 0<s<1/(k–1), s=1/(k–1),and 1/(k–1)<s s_k. For s 1/(k–1) the expansionhas the form where p_s is a log (1/s²)-periodic function, d_s=log (k–1)/log(1/s) is a similarity dimension, A_s and B are constants relatedto the edges and vertices, respectively, of D_s, and r_s is anerror exponent. For s=1/(k–1), the t^1/2-term carries anadditional log t. 1991 Mathematics Subject Classification: 11D25,11G05, 14G05.     

Cohen-Macaulay Properties of Thom-Boardman Strata II: The Defining Ideals of {Sigma}i, j

We consider the Zariski closure of Thom–Boardman strata^{i, j} in the jet space for map-germs (Cⁿ, 0) (C^p, 0). We applythe geometric technique of calculating syzygies to the desingularizationof ^{i, j} constructed by Ronga. We investigate when the normalizationof the coordinate ring of ^{i, j} is rational in the cases i =n – p + 1 or i = j = 1. We also consider when these coordinaterings are normal with rational singularities. 2000 MathematicsSubject Classification 13D02, 14B05, 14M05, 14M12, 58C25 (primary),58K20 (secondary).     

Monodromy and Connectedness

The purpose of this note is initially to present an elementarybut surprising connectedness principle pertaining to the intersectionof a fixed subvariety X of some ambient space Z with anothersubvariety Y which is ‘mobile’ (in the sense ofbeing movable, rather than actually moving). It is via thismobility that monodromy enters the picture, permitting the crucialpassage from ‘relative’ or total-space irreducibilityto ‘absolute’ or fibrewise connectedness (and sometimesirreducibility). A general form of this principle is given inTheorem 2 below. 1991 Mathematics Subject Classification 14C99,15N05.     

Multilevel flexible specification of the production function in health economics

^** Email: grassetti{at}stat.unipd.it^*** Email: e.gori{at}dss.uniud.it^**** Email: simona.minotti{at}unicatt.it Previous studies on hospitals' efficiency often refer to quiterestrictive functional forms for the technology (Aigner et al.,1977, J. Econom., 6, 21–37). In this paper, referringto a study about some hospitals in Lombardy, we formulate convenientcorrectives to a statistical model based on the translogarithmicfunction—the most widely used flexible functional form(Christensen et al., 1973, Rev. Econ. Stat., 55, 28–45).More specifically, in order to take into consideration the hierarchicalstructure of the data (as in Gori et al., 2002, Stat. Appl.,14, 247–275), we propose a multilevel model, ignoringfor the moment the one-side error specification, typical ofstochastic frontier analysis (Aigner et al., 1977, J. Econom.,6, 21–37). Given this simplification, however, we areeasily able to take into account some typical econometric problemsas, e.g. heteroscedasticity. The estimated production functioncan be used to identify the technical inefficiency of hospitals(as already seen in previous works), but also to draw some economicconsiderations about scale elasticity, scale efficiency andoptimal resource allocation of the productive units. We willshow, in fact, that for the translogarithmic specification itis possible to obtain the elasticity of the output (regardingan input) at hospital level as a weighted sum of elasticitiesat ward level. Analogous results can be achieved for scale elasticity,which measures how output changes in response to simultaneousinputs variation. In addition, referring to scale efficiencyand to optimal resource allocation, we will consider the resultsof Ray (1998, J. Prod. Anal., 11, 183–194) to our context.The interpretation of the results is surely an interesting administrativeinstrument for decision makers in order to analyse the productiveconditions of each hospital and its single wards and also todecide the preferable interventions.     

On the rationality of moduli spaces of pointed curves

Motivated by several recent results on the geometry of the modulispaces of stable curves of genus g with n marked points, we determine the birational structureof these spaces for small values of g and n by exploiting suitableplane models of a general curve. More precisely, _g,n is shownto be rational for g = 2 and 1 n 12, g = 3 and 1 n 14, g= 4 and 1 n 15, and g = 5 and 1 n 12     

A generalised Skolem-Mahler-Lech theorem for affine varieties

We give a corrected and strengthened statement and proof ofthe ‘p-adic analytic arc lemma’ in a paper of theauthor (J. London Math. Soc. (2) 73 (2006) 367–379). Weshow that the analytic arc is guaranteed to exist for p 5 andgive a counterexample showing that this sometimes cannot bedone when p = 2. Footnotes 2000 Mathematics Subject Classification 11D45 (primary), 14R10,11D88 (secondary). Received September 16, 2007; revised January 22, 2008; published online March 30, 2008.