2008/06/30 by Roya Beheshti, David Eisenbud
Mathematics · #Algebraic Geometry and Number Theory #Bounded function #Combinatorics #Commutative Algebra and Its Applications #Complete intersection #Degree (music) #Dimension (graph theory) #Discrete mathematics #Disjoint sets #Fiber #Mathematical analysis #Mathematics #Projective variety #Subvariety #Tensor decomposition and applications #Variety (cybernetics) #math.AC #math.AG #msc:13B22 #msc:14B07 #msc:14N05
paper · pdf · doi:10.1112/s0010437x09004503
published as Compositio Math. 146 (2010) 435-456 · Proof of the main theorem simplified and new examples added
arxiv created 2009/08/17 · openalex publication_date 2010/02/02 · arxiv updated 2019/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract Let X be a smooth projective variety of dimension n in P r , and let π : X → P n + c be a general linear projection, with c >0. In this paper we bound the scheme-theoretic complexity of the fibers of π . In his famous work on stable mappings, Mather extended the classical results by showing that the number of distinct points in the fiber is bounded by B := n / c +1, and that, when n is not too large, the degree of the fiber (taking the scheme structure into account) is also bounded by B . A result of Lazarsfeld shows that this fails dramatically for n ≫0. We describe a new invariant of the scheme-theoretic fiber that agrees with the degree in many cases and is always bounded by B . We deduce, for example, that if we write a fiber as the disjoint union of schemes Y ′ and Y ′′ such that Y ′ is the union of the locally complete intersection components of Y , then deg Y ′ +deg Y ′′ red ≤ B . Our method also gives a sharp bound on the subvariety of P r swept out by the l -secant lines of X for any positive integer l , and we discuss a corresponding bound for highly secant linear spaces of higher dimension. These results extend Ran’s ‘dimension +2 secant lemma’.