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Algebraic and Analytic Geometry of Fans
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Memoirs of the American Mathematical Society
1995; 117 pp; softcover
Volume: 115
ISBN-10: 0-8218-2612-3
ISBN-13: 978-0-8218-2612-6
List Price: US$44 Individual Members: US$26.40
Institutional Members: US\$35.20
Order Code: MEMO/115/553

A set which can be defined by systems of polynomial inequalities is called semialgebraic. When such a description is possible locally around every point, by means of analytic inequalities varying with the point, the set is called semianalytic. If one single system of strict inequalities is enough, either globally or locally at every point, the set is called basic. The topic of this work is the relationship between these two notions. Namely, Andradas and Ruiz describe and characterize, both algebraically and geometrically, the obstructions for a basic semianalytic set to be basic semialgebraic. Then they describe a special family of obstructions that suffices to recognize whether or not a basic semianalytic set is basic semialgebraic. Finally, they use the preceding results to discuss the effect on basicness of birational transformations.

• Introduction
• Basic and generally basic sets
• The real spectrum
• Algebraic and analytic tilde operators
• Fans and basic sets
• Algebraic fans and analytic fans
• Prime cones and valuations
• Centers of an algebraic fan
• Henselization of algebraic fans
• A going-down theorem for fans
• Extension of real valuation rings to the henselization
• The amalgamation property
• Algebraic characterization of analytic fans
• Finite coverings associated to a fan
• Geometric characterization of analytic fans
• The fan approximation lemma
• Analyticity and approximation
• Analyticity after birational blowing-down
• References