THE TOPOLOGY OF REAL PROJECTIVE ALGEBRAIC VARIETIES

This article is a survey of the results on Hilbert's 16th problem from 1876 to the present. 1. HARNACK'S THEOREM. The number of branches of a non-singular curve of order m in RP2 does not exceed 1/2(m – 1)(m – 2) + 1. Harnack's method for constructing curves with the greatest number of branches (M-curves). The methods of Hilbert, Brusotti, and Wiman for constructing M-curves. M-curves on a quadric.2. The concepts of roughness and degrees of non-roughness. Brusotti's theorem on the independence of the simplifications (small corrections) of simple double points and cusps.3. The theorems of Petrovskii and Oleinik on algebraic curves and surfaces. The generalization due to Kharlamov.4. Sextic curves in RP2. Cubic and quartic surfaces in RP3.5. An oval of a non-singular curve is said to be odd if it lies inside an odd number of other ovals, and even otherwise. Let A be an M-curve of even order m, and suppose that the number of its even (odd) ovals is P(L). Then P – L (m/2)2mod 8. Broad generalizations were given by Rokhlin and others.6. Some conjectures.

[1]  M. P. Beloch.  Sul numero delle falde delle superficie algebriche , 1949 .

[2]  J. L. Coolidge,et al.  A Treatise on Algebraic Plane Curves , 1932 .

[3]  Annibale Comessatti.  Sulla connessione delle superficie algebriche reali , 1928 .

[4]  H. Whitney.  Elementary Structure of Real Algebraic Varieties , 1957 .

[5]  M. Piazzolla-Beloch.  Sulle superficie del 3o ordine possedenti curve con circuiti concatenati , 1930 .

[6]  Herbert Busemann,et al.  The geometry of geodesics , 1955 .

[7]  M. Atiyah,et al.  The Index of elliptic operators. 3. , 1968 .

[8]  V. I. Arnol'd,et al.  Distribution of ovals of the real plane of algebraic curves, of involutions of four-dimensional smooth manifolds, and the arithmetic of integer-valued quadratic forms , 1971 .

[9]  R. Courant,et al.  Methoden der Mathematischen Physik: Erster Band , 1931 .

[10]  Istvan Fary,et al.  Cohomologie Des Varietes Algebriques , 1957 .

[11]  G. E. Bredon.  Cohomological Aspects of Transformation Groups , 1968 .

[12]  V. Rokhlin.  Congruence modulo 16 in Hilbert's sixteenth problem. II , 1973 .

[13]  Beniamino Segre,et al.  The Non-Singular Cubic Surfaces , 1942 .

[14]  F. Klein.  Ueber Flächen dritter Ordnung , 1873 .

[15]  V. E. Galafassi.  Le questioni di realità come sussidio in altri campi d’indagine , 1960 .

[16]  A. Hurwitz.  Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten , 1891 .

[17]  M. Morse.  Relations between the critical points of a real function of $n$ independent variables , 1925 .

[18]  J. Milnor.  On the Betti numbers of real varieties , 1964 .

[19]  I. Petrowsky,et al.  On the Topology of Real Plane Algebraic Curves , 1938 .

[20]  A. Harnack,et al.  Ueber die Vieltheiligkeit der ebenen algebraischen Curven , 1876 .

[21]  Felix Klein.  Eine neue Relation zwischen den Singularitäten einer algebraischen Curve , 1876 .

[22]  D. Gudkov,et al.  Periodicity of euler characteristic of real algebraic (M — 1)-manifolds , 1973 .

[23]  Topologische Eigenschaften reeller algebraischer Mannigfaltigkeiten , 1950 .

[24]  V. Kharlamov.  A generalized Petrovskii inequality , 1974 .

[25]  D. Hilbert,et al.  Ueber die reellen Züge algebraischer Curven , 1891 .

[26]  M. Hasse.  F. Hirzebruch, Neue Topologische Methoden in der Algebraischen Geometrie. Zweite ergänzte Aufl. VIII + 181 S. Berlin/Göttingen/Heidelberg 1962. Springer-Verlag. Preis brosch. DM 30,80 , 1963 .

[27]  Schlafli.  On the Distribution of Surfaces of the Third Order into Species, in Reference to the Absence or Presence of Singular Points, and the Reality of Their Lines , 1863 .

[28]  Nota di Luigi Brusotti.  Curve generatrici e curve aggregate nella costruzione di curve piane d’ordine assegnato dotate del massimo numero di circuiti , 1916 .

[29]  K. Rohn,et al.  Die Maximalzahl und Anordnung der Ovale bei der ebenen Kurve 6. Ordnung und bei der Fläche 4. Ordnung , 1913 .

[30]  L. Brusotti.  Sulla generazione di curve piane algebriche reali mediante « piccola variazione » di una curva spezzata , 1914 .