计数几何演算法 (The Calculus of Enumerative Geo

计算满足各种条件的代 数曲线和簇的数量是计数代 数几何中的一个基本问题， 而Schubert演算法是解决此 类问题的系统和有效的理论 。这个理论是由Schubert发 展起来的，本书给出了他对 这一理论全面和通俗易懂的 阐述。从一开始，Schubert 演算法理论就吸引了许多伟 大的数学家的注意。例如， Hilbert提出了关于Schubert 演算法的严格论证，作为他 著名的23个问题列表中的 15问题。弦理论的新发展有 助于解决计数几何学中一些 悬而未决的问题，因此重新 燃起了学者们对这一主题的 兴趣。Schubert的这部经典 著作的英译本对于初学者和 计数几何学专家来说都是有 价值和有趣的，读者可以通 过阅读本书了解Schubert如 何思考这些问题以及他如何 提出解决这些问题的方法。 正如Schubert所说，这本书 “应该让读者熟悉一个新的 几何领域的思想、问题和成 果”，并且“应该教授如何处 理一种奇特的演算方法，使 人们能够以简单自然的方式 确定大量的几何数以及奇点 数之间的关系”。

Part I The symbolism of conditions §1 The number of constants of a structure §2 The description of the conditions §3 The dimension of a condition and the level of a system §4 The principle of conservation of numbers §5 The representation of the numbers of con\discretionary- ditions by the symbols of conditions, and computations with these symbols §6 The equations between the elementary conditions of each of the three principal elements Part II The incidence formulae §7 The incidence formulae for points and lines §8 Applications of incidence formulae (I), (II) and (III) to the incidence of a tangent with its point of contact §9 Further eamples for the incidence formulae (I), (II), (III) §10 The remaining incidence formulae §11 Eamples for the incidence formulae (IV) to (XIV) §12 Application of the incidence formulae to systems of principal elements incident with principal elements Part III The coincidence formulae §13 The coincidence formulae of a pair of points and Bezout's theorems §14 Application of the coincidence formulae of \S13 to determine the numbers concerning contacts of planar curves and surfaces §15 The pair of lines and its coincidence conditions §16 Application of the coincidence formulae for pairs of lines to the two ruled families lying on a surface of degree two [23] §17 The pairs of distinct principal elements and the coincidence conditions §18 Derivation of the Cayley-Brill correspondence formula from the general coincidence formulae for pairs of points Part IV The computations of numbers via degeneracies §19 Numbers for structures consisting of finitely many principal elements §20 Numbers for conic sections [30] §21 The reduction of Chasles and Zeuthen [32] §22 Numbers for surfaces of degree two [33] §23 Numbers for cubic planar curves with cusp [34] §24 Numbers for cubic planar curves with double point [34] §25 Numbers for cubic space curves [35] §26 Numbers for planar curves of order four in a fied plane §27 Numbers for the linear congruence [40] §28 Numbers for structures consisting of two lines whose points and planes are projective [41] §29 Numbers for structures consisting of a pencil of planes and a pencil of lines projective to it [41] §30 Numbers for the structure consisting of two projective pencils of lines [41] §31 Numbers for structure consisting of two collinear bundles [42] §32 Numbers for structures consisting of two correlative bundles [42] Part V The multiple coincidences §33 Coincidence of intersection points of a line and a surface [43] §34 The coincidence of multiple points on a line [48] §35 The coincidence of multiple lines of a pencil of lines [48] §36 Singularities of the generic line comple [49] Part VI The theory of characteristics §37 The problem of characteristics for an arbitrary structure Gamma §38 The problem of characteristics for the conic section [51] §39 Derivation and application of the characteristic formulae for the structure consisting of a line and a point on it [52] §40 Derivation and application of the characteristic formula for the pencil of lines [52] §41 Derivation and application of the characteristic formula for the structure consisting of a line, a point on that line, and a plane through that line [52] §42 The theory of characteristics of the structure consisting of a line and n points on it [53] §43 Computation of the numbers for multiple secants of the intersection curve of two surfaces §44 Theory of characteristics of the structure consisting of a pencil of lines and n lines in it. Application to congruences common to two complees