Theory Of Algebraic Surfaces Research Articles

On the boundedness of the denominators in the Zariski decomposition on surfaces

Abstract Zariski decompositions play an important role in the theory of algebraic surfaces. For making geometric use of the decomposition of a given divisor, one needs to pass to a multiple of the divisor in order to clear denominators. It is therefore an intriguing question whether the surface has a “universal denominator” that can be used to simultaneously clear denominators in all Zariski decompositions on the surface. We prove in this paper that, somewhat surprisingly, this condition of bounded Zariski denominators is equivalent to the bounded negativity of curves that is addressed in the Bounded Negativity Conjecture. Furthermore, we provide explicit bounds for Zariski denominators and negativity of curves in terms of each other.

Diametral theory of algebraic surfaces and geometric theory of invariants of groups generated by reflections. III

Diametral theory of algebraic surfaces and geometric theory of invariants of groups generated by reflections. III

Calculus on Arithmetic Surfaces

In [A2] and [A3], Arakelov introduces an intersection calculus for arithmetic surfaces, that is, for stable models of curves over a number field. In this paper we intend to show that his intersection product has a lot of useful properties. More precisely, we show that the following properties from the theory of algebraic surfaces have an analogue in our situation:

The Hyperplane Sections of Arithmetically Normal Varieties

In [S], showed that a generic hyperplane section, and also almost every hyperplane section, of a normal variety of dimension d > 1 is normal. In [Nk], Nakai considered arithmetically normal varieties and wrote: we cannot expect, in general, that the hyperplane section possesses also this property. This is, of course, the expression of a conjecture. Below give (counter-) examples (of varieties of dimension d > 1) showing that the conjecture is correct: if F, G are plane curves without singularities, then, except in the case that F is a line or a conic and G is a line or a conic, the Segre embedding of F X G yields an example. Of these examples, the cases of a line times a cubic, a line times a quartic, a conic times a cubic, and a cubic times a cubic are relatively easy to get; the others require some deeper points from the theory of algebraic surfaces (one shows that any arithmetically normal irregular surface free of singularities yields an example). In [Nk], Nakai developed a necessary and sufficient condition for a generic hyperplane section of an arithmetically normal variety to be arithmetically normal. We also develop such a condition, namely, a generic hyperplane section of an arithmetically normal variety V of dimension d > 1 in n space is arithmetically normal if and only if the cohomological dimensionl of V (i. e., of its associated prime ideal) is less than n 1; hence if d = 2, if and only if V (i. e., its prime ideal) is perfect. But did not see how to resolve the issue of whether this necessary and sufficient condition always obtains without giving a counter-example. As were looking for an arithmetically normal variety V with imperfect associated prime ideal, consulted the paper [G2] of GrPbner, who gives (new) examples of imperfect prime ideals; but none of these corresponds to an arithmetically normal variety, as one may verify directly. Our surfaces F X G (except in the cases line times line, etc.) yield arithmetically normal imperfect ideals.

Introduction to the Problem of Minimal Models in the Theory of Algebraic Surfaces. By O. Zariski. Pp. 90. $2. 1958. (The Mathematical Society of Japan)

Introduction to the Problem of Minimal Models in the Theory of Algebraic Surfaces. By O. Zariski. Pp. 90. $2. 1958. (The Mathematical Society of Japan)

The Problem of Minimal Models in the Theory of Algebraic Surfaces

The Problem of Minimal Models in the Theory of Algebraic Surfaces

Fibre Systems of Jacobian Varieties

The method of Picard and Poincare in studying an algebraic surface V consists in applying the theory of curves to the curves of an auxililary linear pencil {Cj on V. The same method was employed later by Lefschetz in his theory of algebraic surfaces (cf. [19], Chapters 6-7). If we examine closely their method, we can see that they considered implicitly a certain variety cJ which the author proposes to call the Teron variety of V associated with {CJ. In fact Neron was the first who considered g explicitly in his algebraic proof of the theorem of the base [12]. The variety #J is the graph of the correspondence u ->Ju between u and the Jacobian variety JX of Cu. Here, we restrict our attention to such linear pencils whose members are all irreducible and whose general members are nonsingular. If V does not carry any multiple curve, we can always find such a linear pencil. Also, if V is nonsingular, we can assume that singular members of the pencil are curves with ordinary double points. Now, the main part of the paper is devoted to determining the degenerate fibres of the fibre system {u X J,} on # at those finite number of values of u where Cu become singular. We note that in Neron's case such degenerate fibres are not considered explicitly. The same thing can be said about Chow's investigations on Abelian varieties over function fields [4]. However, in some problems in algebraic geometry it becomes necessary to consider those degenerate fibres and also the behavior of fibres along the degenerate fibres. We shall show that the degenerate fibres are certain completions of the generalized Jacobian varieties of the singular curves in the sense of Rosenlicht [15]. The singular locus of #J is contained in the union of singular loci of degenerate fibres. Also we can define in a natural way a birational map p from V into g., and we can show that p gives isomorphisms of the Albanese varieties and of the spaces of linear differential forms of the first kind of V and l. This result has already been applied to show that the dimension of the Albanese variety of an arbitrary

Independent integral bases and a characterization of regular surfaces

Introduction. The fundamental paper Theorie der algebraischen Funktionen einer Verdnderlichen of Dedekind and Weber opened a wide field of research. The methods which these authors brought into play in the study of algebraic functions of one variable have lent themselves readily, from the conceptual point of view at least, to generalization to the case of several variables. However, even if one restricts himself to the case of algebraic functions of two variables he finds a sharp line of demarcation in the analogy with the one variable case when he attacks problems of enumeration such as the problem of Riemann-Roch. Several years ago Zariski made the conjecture that the analogy between the theory of algebraic curves and the theory of algebraic surfaces would extend much further in the case of regular surfaces, that is surfaces whose arithmetic and geometric genera coincide. The objective of this paper is to establish the truth of this conjecture and to show, that this fact is in a sense characteristic of regular surfaces. A tool which proved to be most useful to Dedekind and Weber in deriving the Riemann-Roch theorem as well as other allied results was furnished by their so-called normal bases, the construction of which we shall describe briefly below. Let z be a field of algebraic functions of one variable x cver a field of constants K, which is assumed to be maximally algebraic in z and of characteristic zero. The field z is then an algebraic extension of K(x) of finite relative degree v. Let o denote the ring of integral functions of x in z and o' the ring of integral functions of l/x in Z. If X is an element of o, the of w (in. symbols exp co) is defined to be the smallest integer r such that w/xr is an element of o' (that is cO/xrCo', Lo/xr-l1o'). A set of functions X1, X2, , X, is constructed as follows. Let X1 = 1, and let X2 be an element in o of lowest exponent r2, such that X2 does not satisfy a congruence of the form X2=CX1(0X) where ceK. If Xl, X2, , Xi-, have been selected, take Xi to be an element in o of lowest exponent which does not satisfy a congruence of the form Xi -c1Xi+ + +c-ix i-i1(ox), c jeK, j=1, , i-1. Dedekind and Weber show that this construction leads to a set of v(-11: K(x) 1) fulnctions Xi, X2, . ., X, in o which have the following properties:

Principles of Geometry. V. Analytical principles of the theory curves. Pp. x, 247. 15s. VI. Introduction to the theory of algebraic surfaces and higher loci. Pp. ix, 308. 17s. 6d. By Baker H. F.. 1933. (Cambridge)

Principles of Geometry. V. Analytical principles of the theory curves. Pp. x, 247. 15s. VI. Introduction to the theory of algebraic surfaces and higher loci. Pp. ix, 308. 17s. 6d. By Baker H. F.. 1933. (Cambridge)

Isolated singular points in the theory of algebraic surfaces

If F(x0, x1, x2, x3) = 0 is the equation of a surface in space of three dimensions which has an ordinary isolated s-ple point O, then by means of the substitutionswhere Φ0 = 0, Φ1 = 0, …, Φr = 0 are the equations of r + 1 linearly independent surfaces passing simply through O, F is transformed into a surface F′ in [r], on which to the point O of F there corresponds a simple curve γ. The points of γ arise from the points of F in the first neighbourhood of O, and in this simple case the genus of γ is ½ (s − 1) (s − 2). In the study of properties which are common to all members of an infinite family of birationally equivalent surfaces no distinction is made between O and γ, O being regarded as a curve which has become infinitesimal on the particular surface of the family in question.

On Some Recent Advances in the Theory of Algebraic Surfaces

On Some Recent Advances in the Theory of Algebraic Surfaces

Societies and Academies

PARIS. Academy of Sciences, October 9.—M. van Tieghem in the chair.—On the elastic equilibrium of a rectangular plate, by M. Maurice Lévy.—Some remarks on double integrals of the second species in the theory of algebraic surfaces, by M. Emile Picard.—On a modification of Bessel's method for calculating occultations, by M. L. Cruls.

Societies and Academies

PARIS. Academy of Sciences, October 24.—M. Wolf in the chair.—On double integrals of the second species in the theory of algebraic surfaces, by M. Émile Picard.—Properties of calcium, by M. Moissan. The pure crystallised calcium whose properties are given in this paper, was prepared by the method already described in NATURE. The melting point, determined by a thermo-couple, was found to be 760° C.