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## Über dieses Buch

The aim of the present monograph is to give a systematic exposition of the theory of algebraic surfaces emphasizing the interrelations between the various aspects of the theory: algebro-geometric, topological and transcendental. To achieve this aim, and still remain inside the limits of the allotted space, it was necessary to confine the exposition to topics which are absolutely fundamental. The present work therefore makes no claim to completeness, but it does, however, cover most of the central points of the theory. A presentation of the theory of surfaces, to be effective at all, must above all give the typical methods of proof used in the theory and their underlying ideas. It is especially true of algebraic geometry that in this domain the methods employed are at least as important as the results. The author has therefore avoided, as much as possible, purely formal accounts of results. The proofs given are of necessity condensed, for reasons of space, but no attempt has been made to condense them beyond the point of intelligibility. In many instances, due to exigencies of simplicity and rigor, the proofs given in the text differ, to a greater or less extent, from the proofs given in the original papers.

## Inhaltsverzeichnis

### Chapter I. Theory and Reduction of Singularities

Abstract
Let x1, x2,…, xr+1 be homogeneous point coördinates in a complex projective γ-dimensional space Sr. An algebraic variety V in S γ is the locus of point (x) satisfiying a system of algebraic equations,
$${f_1}\left( {{x_{1,}} \ldots ,{x_\gamma }_{ + 1}} \right) = 0, \ldots ,{f_n}\left( {{x_1}, \ldots ,{x_{\gamma + 1}}} \right) = 0$$
(1)
, where f1, f2,…, fn are homogeneous polynomials. If ϕ is a homogenous polynomial in the x’s which vanishes at all the common zeros of f1, …, fn, i. e. at every point of V, we say briefly that ϕ vanishes (ϕ= 0) on V. The variety V is irreducible, if from ϕ ψ = 0 on V it follows necessarily that one at least of the polynomials ϕ, ψ vanishes on V. In the language of the theory of ideals this definition can be formulated as follows: V is irreducible if the homogeneous polynomial ideal (f1,…fn) (H-ideal) is a primary ideal1 (Macaulay, a, p. 33; van der Waerden, a2, p. 54). From the theorem of Hilbert-Netto (Macaulay, a, p. 48; van der Waerden, a2, p. 11 ) it follows then that either ϕϱ or ψϱ is a member of (f1,…, fn), where ϱ is a convenient integer.
Oscar Zariski

### Chapter II. Linear Systems of Curves

Abstract
In the sequel we shall consider only algebraic surfaces without singularities in Sr, or with ordinary singularities in S3, obtained from the former surfaces by a generic projection. The points of the nodal curve, with the exception of the cuspidal points, should be considered as pairs of distinct points of the surface. It clearly appears from the projection that the cuspidal points are the branch points of the doubly covered nodal curve.
Joseph Lipman

### Chapter III. Adjoint Systems and the Theory of Invariants

Abstract
If $${\left| C \right|_H}$$ is a complete linear system of plane curves of order n, defined by a set $$H = \sum O _i^{{8_i}}$$ of base points, the dimension $${\gamma _n}of{\left| C \right|_H}$$ is given, for n sufficiently large, say for $$n \geqslant l$$ by the postulation formula (Cayley, Noether) or by the characteristics formula (Hilbert): $${\gamma _n} = \left( {n\mathop { + 2}\limits_2 } \right) - k - 1$$ where $$k\underline{\underline < } \sum {\frac{{{S_i}\left( {{S_i} + 1} \right)}}{2}}$$ is an integer independent of n. If the proximity inequalities are satisfied, then $$k = \sum {\frac{{{s_i}\left( {{s_i} + 1} \right)}}{2}}$$. For n < l the effective dimension $$\gamma \# {\rm{x2032}}{;_{\rm{n}}}\} {\rm{of}}{\left| {\rm{C}} \right|_{\rm{H}}}$$ is greater than its virtual dimension γn evaluated according to the postulation formula. The difference s = γ n - γ n is called the superabundance of $${\left| C \right|_H}$$. A system is superabundant if s>0, regular if s = 0.
David Mumford

### Chapter IV. The Arithmetic Genus and the Generalized Theorem of Riemann-Roch

Abstract
Let F be a surface of order n with ordinary singularities in S3. The effective dimension of the complete system $$\left| {{\varphi _{n - 4}}} \right|$$ of adjoint sufaces of F of order $$n - 4{\rm{ }}is\,{P_g} - 1$$. Cayley (1) and Noether (2) have first shown by the examples that the virtual dimesion of $$\left| {{\varphi _{n - 4}}} \right|$$ evaluated according to the postulation formula (III, 2) may be less than P - 1. Thus, for instance, if F is ruled surface of genus P g , it is found (Cayley, 1) that $${P_a} = - P,{\text{ while }}{{\text{P}}_g} = 0$$, because in this case $$\left| {{\varphi _{n - 3}}} \right|$$ and hence a fortiriori $$\left| {{\varphi _{n - 4}}} \right|$$ does not exist (III, 3). The virtual dimension of $$\left| {{\varphi _{n - 4}}} \right|$$ incresed by 1 is called the arithmetic genus of F and is denoted by Pa (Noether, 2). From the well known fact that the adjoint curves of order ≧n - 3 of a plane irreducible curve of order n form a regular system, it follows in view of III, 2, formula (2′), that $${P_g}\underline{\underline > } {P_a}$$ and that
$${P_g} - {P_a} = {\omega _{n - 3}} + {\omega _{n - 2}} + \cdots$$
(1)
, where $${\omega _{n - 3 + i}}$$ is the deficiency of the system of curves cut out by $$\left| {{\varphi _{n - 3 + i}}} \right|$$ on a generic plane. The difference $$q = {P_g} - {P_a}$$ is called the irregularity of F and F is said to be a regular or an irregular surface according as $$q = 0{\text{ }}or{\text{ }}q > 0$$.
David Mumford

### Chapter V. Continuous Non-linear Systems

Abstract
In II, 1 we have accepted for temporary purposes a definition of an algebraic system Σ of curves C on a surface F, as a system cut out on F by an algebraic system Σ′ of hypersurfaces. It will serve as a preliminary clarifying remark if we point out immediately why that definition is not sufficiently general. The base loci of the given system Σ′ of hypersurfaces determine a com­plete linear system of hypersurfaces of the same order as the hyper­surfaces of Σ′. This linear system cuts out on F a linear system of curves of the same order as the curves C and containing Σ. Hence Σ is totally contained in a linear system of curves. However—and this is a fundamental point of the theory, which will be discussed in section 3 of this chap­ter—there exist surfaces (notably, irregular surfaces) which carry algebraic systems of curves not contained in linear systems. The simplest example is given by surfaces carrying an irrational pencil Σ of curves. Obviously, such a pencil (supposing for simplicity that the curves of the pencil are irreducible) cannot be contained in a linear system of dimension r, where r is necessarily ≧2, because the curves of the pencil are of virtual degree zero (II, 1). If the pencil is reducible, it is seen immediately that its curves are composed of the curves of another irrational pencil, and the statement that Σ is not contained in a linear system is essentially equivalent to the statement to the statement that an irrational involution of sets of points on an algebraic curve cannot be contained in a linear series.
David Mumford

### Chapter VI. Topological Properties of Algebaraic Surfaces

Abstract
In the following exposition the fundamental notions of Analysis Situs will be assumed known. The terms and notations are as in Lefschetz, e. For the convenience of the reader we give a list of those most frequently used in the sequel.
David Mumford

### Chapter VII. Simple and Double Integrals on an Algebraic Surface

Abstract
a. Simple integrals. A simple integral, or integral of total differential, attached to an algebraic surface F $$f\left( {x,y,z} \right) = 0$$ is an integral of the form
$$I = \int\limits_{\left( {{x_0},{y_0},{z_0}} \right)}^{\left( {x,y,z} \right)} Q \left( {x,y,z} \right)dx + R\left( {x,y,z} \right)dy$$
(1)
where Q and R are rational functions of x, y, z satisfying the integrability condition $$\frac{{\partial Q}}{{\partial y}} = \frac{{\partial R}}{{\partial x}}$$, the derivatives being evaluated by considering z as an implicit function of x and y. These integrals have been introduced by Picard (a, 2, 3, 4). The integral I may possess on F either polar or logarithmic singularities, and their locus is referred to respectively as the polar or the logarithmic curve of I. The integral is of the first, second or third kind according as it possesses no singularities, polar singularities only, or logarithmic singularities. An integral I of the first or second kind is also characterized by the condition that $$\int {d{\text{ }}I = 0}$$ for any 1-cycle Г1 on F which is homologous to zero. If I is any simple integral and if Г1 is any 1-cycle on F, the integral $$\int\limits_{{\Gamma _1}} {d{\text{ }}I}$$ is called the period of I relative to Г1 if Г1 ≁ 0 on F, a logarithmic period if Г1 ∽ 0. A simple integral I without periods (i. e. whose periods all vanish) is a constant, a rational function or a logarithmo-rational function of F, according as I is of the first, second or third kind.
David Mumford

### Chapter VIII. Branch Curves of Multiple Planes and Continuous Systems of Plane Algebraic Curves

Abstract
Let z be a k-valued algebraic function of two complex variables x and y, defined by an irreducible algebraic equation,
$$F(x,y,z){\rm{ }} = {\rm{ }}0$$
(1)
. The branch curve f,
$$f(x,y){\rm{ }} = {\rm{ }}0$$
(2)
, of the function z is found by eliminating z between F = 0 and $$\partial F/\partial z = 0$$ and by neglecting in the resultant certain factors which correspond to multiple curves of the surface F = 0 (apparent branch curves) The definition of f may be rendered exact by assuming that: (a) the polynomial f contains no multiple factors; (b) the curve f is the locus of the effective branch points (x1, y), (x2, y),…,(xn, y) of the function z = z(x,y), for y fixed, and of the lines y = c = const. such that y = c is an effective branch point of z if x is fixed and generic. It may be necessary to include the line at infinity of the projective plane (x,y) in the branch curve. However, we may always choose the coördinates x and y in such a manner that the line at infinity does not belong to the branch curve.
Sheeram Shankar Abhyankar, David Mumford

### Backmatter

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