What Determines an Algebraic Variety? (eBook)

Janos Kollar, Max Lieblich, Martin Olsson, Will Sawin (Autoren)

eBook Download: PDF

2023
240 Seiten
Princeton University Press (Verlag)
978-0-691-24683-3 (ISBN)

Infos

A pioneering new nonlinear approach to a fundamental question in algebraic geometryOne of the crowning achievements of nineteenth-century mathematics was the proof that the geometry of lines in space uniquely determines the Cartesian coordinates, up to a linear ambiguity. What Determines an Algebraic Variety? develops a nonlinear version of this theory, offering the first nonlinear generalization of the seminal work of Veblen and Young in a century. While the book uses cutting-edge techniques, the statements of its theorems would have been understandable a century ago; despite this, the results are totally unexpected. Putting geometry first in algebraic geometry, the book provides a new perspective on a classical theorem of fundamental importance to a wide range of fields in mathematics.Starting with basic observations, the book shows how to read off various properties of a variety from its geometry. The results get stronger as the dimension increases. The main result then says that a normal projective variety of dimension at least 4 over a field of characteristic 0 is completely determined by its Zariski topological space. There are many open questions in dimensions 2 and 3, and in positive characteristic.

Produkt-Details

• Erscheint lt. Verlag: 25.7.2023
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Zusatzinfo: 4 b/w illus.
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Schlagworte: Abelian • Algebraic • algebraically • Algebraically closed • Ample • Ample divisor • assume • Bundle • Cartier • closure • complete • Component • Curve • definable • degree • Denote • Dense • dim • Dimension • dimensional • Disjoint • Divisor • Divisorial • Effective • Elements • Equivalence • Equivalent • extension • fiber • Field • Finite • Finitely • Function • Geometrically • Hilbertian • Infinite • Integral • Intersection • Irreducible • isomorphism • line • Linear • locus • map • Morphism • Nonempty • pencil • Points • Positive • PROJECTIVE • quasi • Rational • results • Scheme • section • Separable • sheaf • smooth • space • Structure • Subset • Suffices • Theorem • Topological • Torsion • Unique • variety • Weil • Zariski • Zero
• ISBN-10: 0-691-24683-1 / 0691246831
• ISBN-13: 978-0-691-24683-3 / 9780691246833