• Preface (7)
• Contents (13)
• 1 Prerequisites (17)
• 2 Algebraic Sets, Morphisms, and Rational Maps (28)
  • Review of topology (28)
  • The correspondences V and I (31)
  • Morphisms (37)
  • Rational maps (40)
  • Projective algebraic sets (46)
  • Rational maps and birational equivalence (51)
  • Complements and exercises (59)
• 3 Geometric Properties of Algebraic Varieties (67)
  • Tangent space, singularities and dimension (67)
  • Independence of polynomials. Essential parameters (78)
  • Dimension of a projective variety (84)
  • Order of a projective variety, tangent cone and multiplicity (88)
• 4 Rudiments of Elimination Theory (99)
  • Resultant of two polynomials (99)
  • Bézout's theorem for plane curves (105)
  • More on intersection multiplicity (106)
  • Elimination of several variables (115)
  • Bézout's theorem (118)
• 5 Hypersurfaces in Projective Space (122)
  • Generalities on hypersurfaces (122)
  • Multiple points of a hypersurface (124)
  • Algebraic envelopes (131)
  • Polarity with respect to a hypersurface (135)
  • Quadrics in projective space (142)
  • Complements on polars (150)
  • Plane curves (156)
  • Surfaces in P3 (165)
• 6 Linear Systems (182)
  • Linear systems of hypersurfaces (182)
  • Hypersurfaces of a linear system that satisfy given conditions (184)
  • Base points of a linear system (186)
  • Jacobian loci (193)
  • Simple, composite, and reducible linear systems (198)
  • Rational mappings (201)
  • Projections and Veronese varieties (205)
  • Blow-ups (208)
• 7 Algebraic Curves (213)
  • Generalities (213)
  • The genus of an algebraic curve (217)
  • Curves on a quadric (228)
  • Rational curves (234)
  • Exercises on rational curves (241)
• 8 Linear Series on Algebraic Curves (254)
  • Divisors on an algebraic curve with ordinary singularities (255)
  • Linear series (262)
  • Linear equivalence (264)
  • Projective image of linear series (267)
  • Special linear series (272)
  • Adjoints and the Riemann–Roch theorem (277)
  • Properties of the canonical series and canonical curves (286)
  • Some results on algebraic correspondences between two curves (290)
  • Some remarks regarding moduli (293)
  • Complements and exercises (300)
• 9 Cremona Transformations (308)
  • Quadratic transformations between planes (308)
  • Resolution of the singularities of a plane algebraic curve (313)
  • Cremona transformations between planes (323)
  • Cremona transformations between projective spaces of dimension 3 (333)
  • Exercises (338)
• 10 Rational Surfaces (356)
  • Planar representation of rational surfaces (356)
  • Linearly normal surfaces and their projections (365)
  • Surfaces of minimal order (370)
  • The conics of a plane as points of P5 and the Veronese surface (376)
  • Complements and exercises (380)
• 11 Segre Varieties (402)
  • The product of two projective lines (402)
  • Segre morphism and Segre varieties (405)
  • Segre product of varieties (408)
  • Examples and exercises (411)
• 12 Grassmann Varieties (415)
  • The lines of P3 as points of a quadric in P5 (415)
  • Complexes of lines in P3 (419)
  • Congruences of lines in P3 (423)
  • Ruled surfaces in P3 (424)
  • Grassmann coordinates and Grassmann varieties (430)
  • Further properties of G(1,n) and applications (438)
• 13 Supplementary Exercises (449)
  • Miscellaneous exercises (449)
  • Further problems (470)
  • Exercises on linear series on curves (473)
• Bibliography (483)
• Index (491)