Primitive linear series on curves

In this paper we study a new numerical invariant ℓ of curvesC which is related to the primitive linear series onC. (Primitive series—defined below—are the essential complete and special linear series onC.) The curves with ℓ≤3 are classified, and it is shown that for a given value of ℓ the curve is a double covering if its genus is sufficiently high. The main tool are dimension theorems of H. Martens-Mumford-type for the varieties of special divisors ofC, and we prove two refinements of these theorems.     

Non complete linear series on curves

Abstract The paper is concerned with some properties of linear series on smooth plane curves; in fact, we study mainly the case of cubic curves. The main result describes the growth of the dimension of non complete linear series, generalizing to cubics a well known result of Gieseker, about linear series on the projective lines. Keywords: Curves, Linear series Mathematics Subject Classification (2000): 14Q05     

On the Halphen transform of algebraic space curves

The Halphen transform of a plane curve is the curve obtained by intersecting the tangent lines of the curve with the corresponding polar lines with respect to some conic. This transform was introduced by Halphen as a branch desingularization method in [5 Halphen, G. H. (1876). Sur une série de courbes analogues aux développées. J. Math. Pures Appl. 3e série, tome 2:87–144. [Google Scholar]] and has also been studied in [2 Coolidge, J. L. (2004). A Treatise on Algebraic Plane Curves. New York: Dover Publications, Inc., 1959, xxiv+513 pp. [Google Scholar], 8 Josse, A. (1995). Transformation d’Halphen. (French) [The Halphen transform]. Commun. Algebra 23(12):4343–4364.[Taylor & Francis Online] , [Google Scholar]]. We extend this notion to the Halphen transform of a space curve and study several of its properties (birationality, degree, rank, class, desingularization).     

Arithmetic inflection formulae for linear series on hyperelliptic curves

Over the complex numbers, Plücker's formula computes the number of inflection points of a linear series of fixed degree and projective dimension on an algebraic curve of fixed genus. Here, we explore the geometric meaning of a natural analog of Plücker's formula and its constituent local indices in ${\mathbb{A}}^1$-homotopy theory for certain linear series on hyperelliptic curves defined over an arbitrary field.     

Completeness and non-speciality of linear series on space curves and deficiency of the corresponding linear series on normalization

Summary Let be a curve ofP ^r (r3) of degree d, C its normalization and , I() a saturated, homogeneous ideal of k[X₀, ...,X _r]. In this paper we show that, if N 0 is an integer such that, for nN, the linear series cut out on by the hypersurfaces of degree n is complete and non-special, then the deficiency of the linear series cut out on C by the hypersurfaces ofA _n,forn>N, is independent ofn and can be explicitly calculated;this is the case, for instance, whenN=d–r+1, and when N=n_i –r–1 (under suitable conditions) if is a component of the complete intersection of r–1 hypersurfaces of degrees n_i.Under financial support from the N.S.E.R.C. of Canada, the italian M.P.I. and the N.A.T.O. Fellowships Scheme Programme.The author wishes to thank R.Lazarsfeld for advice and the Curves Seminar group at Queen's, in particular A. V.Geramita and E.Davis, for fruitful and stimulating discussions on this subject.     

Limit linear series: Basic theory

In this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves, curves of compact type. The technically much simpler special case of 1-dimensional series was developed by Beauville [2], Knudsen [21–23], Harris and Mumford [17], in the guise of “admissible covers”. It has proved very useful for studying the Moduli space of curves (the above papers and Harris [16]) and the simplest sorts of Weierstrass points (Diaz [4]). With our extended tools we are able to prove, for example, that:

1. The Moduli spaceM _g of curves of genusg has general type forg≧24, and has Kodaira dimension ≧1 forg=23, extending and simplifying the work of Harris and Mumford [17] and Harris [16].
2. Given a Weierstrass semigroup Γ of genusg and weightw≦g/2 (and in a somewhat more general case) there exists at least one component of the subvariety ofM _g of curves possessing a Weierstrass point of semigroup Γ which has the “expected” dimension 3g-2?w (and in particular, this set is not empty).
3. The fundamental group of the space of smooth genusg curves having distinct “ordinary” Weierstrass points acts on the Weierstrass points by monodromy as the full symmetric group.
4. Ifr andd are chosen so that $$\rho : = g - (r + 1)(g - d + r) = 0,$$ then the general curve of genusg has a certain finite number ofg _d ^r’ s [15, 20]. We show that the family of all these, allowing the curve to vary among general curves, is irreducible, so that the monodromy of this family acts transitively. If4=1, we show further that the monodromy acts as the full symmetric group.
5. Ifr andd are chosen so that $$\rho = - 1,$$ then the subvariety ofM _g consisting of curves posessing ag _d ^r has exactly one irreducible component of codimension 1.
6. For anyr, g, d such that ρ≦0, the subvariety ofM _g consisting of curves possessing ag _d ^r has at least one irreducible component of codimension—ρ so long as $$\rho \geqq \left\{ \begin{gathered} - g + r + 3 (r odd) \hfill \\ - \frac{r}{{r + 2}}g + r + 3 (r even). \hfill \\ \end{gathered} \right.$$

In this paper we present the basic theory of “limit linear series” necessary for proving these results. The results themselves will be taken up in our forthcoming papers [8-12]. Simpler applications, not requiring the tools developed in this paper but perhaps clarified by them, have already been given in our papers [5-7].     

A remark on Weierstrass points and linear series on multiple covering curves

Here we study multiple coverings of rational and irational curves. We give a theorem about the non-gap sequence on m-gonal curves. We then study general irrational covering f ： X→ C, and say when h^0（X, f^＊（L）） = h^0（C,L） for L line bundle on C.     

Dimension counts for limit linear series on curves not of compact type

We first prove a generalized Brill–Noether theorem for linear series with prescribed multivanishing sequences on smooth curves. We then apply this theorem to prove that spaces of limit linear series have the expected dimension for curves of pseudocompact type, whenever the gluing conditions in the definition of limit linear series impose the maximal codimension. Finally, we investigate these gluing conditions in specific families of curves, showing expected dimension in several cases, each with different behavior. One of these families sheds new light on the work of Cools, Draisma, Payne and Robeva in tropical Brill–Noether theory, and suggests directions of further work in that setting.     

On linear series on general -gonal projective curves

Let be a general -gonal curve of genus . Here we prove a strong upper bound for the dimension of linear series on , i.e. we prove that .

     

Towards a theory of classification

The well-known difficulties arising in a classification which is not set-theoretically trivial—involving what is sometimes called a non-smooth quotient—have been overcome in a striking way in the theory of operator algebras by the use of what might be called a classification functor—the very existence of which is already a surprise. Here the notion of such a functor is developed abstractly, and a number of examples are considered (including those which have arisen for various classes of operator algebras).     

Towards a theory of associatives

In this paper the theorem on the representation of partially conjugate associative n-ary operations by binary and unary operations is proved.Translated from Matematicheskii Zametki, Vol. 11, No. 5, pp. 545–554, May, 1972.     

Towards a unified bifurcation theory

Bifurcation theories for the instability of slowly evolving systems have been developed in various disciplines, and a first step is here taken towards some desirable unification. A modern account of the authors' general branching theory for discrete systems is first presented, some new features being the introduction of principal imperfections and the delineation of the important semi-symmetric points of bifurcation. This theory, embedded in a perturbation approach ideal for quantitative analysis, is complementary to the far-reaching qualitative catastrophe theory of René Thom which offers a profound topological classification of instability phenomena. For this reason, we present here a detailed correlation of the two theories. Also presented in the paper is a survey of some fields of application ranging from classical fields such as hydrodynamics, through thermodynamics, crystallography and cosmology, to the newer domains of biology and psychology.     

Geometry of curves with exceptional secant planes: linear series along the general curve

We study linear series on a general curve of genus g, whose images are exceptional with regard to their secant planes. Working in the framework of an extension of Brill?CNoether theory to pairs of linear series, we prove that a general curve has no linear series with exceptional secant planes, in a very precise sense, whenever the total number of series is finite. Next, we partially solve the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes associated with the family, by evaluating our hypothetical formula along judiciously-chosen test families. As an application, we compute the number of linear series with exceptional secant planes on a general curve equipped with a one-dimensional family of linear series. We pay special attention to the extremal case of d-secant (d ? 2)-planes to (2d ? 1)-dimensional series, which appears in the study of Hilbert schemes of points on surfaces. In that case, our formula may be rewritten in terms of hypergeometric series, which allows us both to prove that it is nonzero and to deduce its asymptotics in d.     

Halphen pencils on quartic threefolds

For any smooth quartic threefold in P⁴ we classify pencils on it whose general element is an irreducible surface birational to a surface of Kodaira dimension zero.     

Towards a self-consistent theory of volatility

In this paper, we propose a new theory for the formation of volatility which takes into account the influence of option hedging on the assets price dynamics. By analogy with statistical mechanics, we build a self-consistent equation for the volatility, we show it is well-posed and we explain how it can be solved.     

Towards a new theory of confirmation

Any sequence of events can be “explained” by any of an infinite number of hypotheses. Popper describes the “logic of discovery” as a process of choosing from a hierarchy of hypotheses the first hypothesis which is not at variance with the observed facts. Blum and Blum formalized these hierarchies of hypotheses as hierarchies of infinite binary sequences and imposed on them certain decidability conditions. In this paper we also consider hierarchies of infinite binary sequences but we impose only the most elementary Bayesian considerations. We use the structure of such hierarchies to define “confirmation”. We then suggest a definition of probability based on the amount of confirmation a particular hypothesis (i.e. pattern) has received. We show that hypothesis confirmation alone is a sound basis for determining probabilities and in particular that Carnap’s logical and empirical criteria for determining probabilities are consequences of the confirmation criterion in appropriate limiting cases.