On degenerate sections of vector bundles
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Mathematische Zeitschrift
On degenerate sections of vector bundles Dennis Tseng1 Received: 5 November 2017 / Accepted: 15 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We consider the locus of sections of a vector bundle on a projective scheme that vanish in higher dimension than expected. We will find the largest components of this locus asymptotically, after applying a high enough twist to the vector bundle. We will also give an interpretation in terms of a limit in the Grothendieck ring of varieties.
1 Introduction We will work throughout over an algebraically closed field K of arbitrary characteristic. Let X be an irreducible projective scheme and V be a globally generated vector bundle on X . A general section s ∈ H 0 (V ) either vanishes in codimension exactly rank(V ) or is nonvanishing. Considering H 0 (V ) as an affine space, there is a closed locus D(V ) ⊂ H 0 (V ) consisting of sections that vanish in codimension less than rank(V ). We are interested in basic questions about this locus, for example: Problem 1.1 What is the dimension of D(V )? What are the components, and what can we say about them? The purpose of this paper is to give a clean answer after a twist of V by a high tensor power of O X (1). For example, we show Theorem 1.2 There exists N0 such that for all N ≥ N0 , any section s ∈ H 0 (V (N )) in any component of D(V (N )) of largest dimension vanishes on some subscheme of X of codimension rank(V ) − 1 and of minimal degree among all subschemes of codimension rank(V ) − 1 in X . Given Theorem 1.2, one may then ask for next largest components of D(V (N )) and expect that they are given by sections that vanish on some subscheme of second largest degree among all varieties of codimension rank(V )−1. Then, the next largest should correspond to varieties of third largest degree and so on.
This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 1144152.
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Dennis Tseng [email protected] Harvard University, Cambridge, MA 02138, USA
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D. Tseng
We will show that this is in fact true (see Theorem 3.1) and will use the Hilbert polynomial instead of degree as a finer invariant to distinguish between the dimensions of the various components of D(V (N )). In particular, the number of components of D(V (N )) will necessarily approach infinity as N → ∞, but we will only have fine control of the dimensions of its large dimensional components. Finally, one can define D(V (N ), a) as the sections that vanish in codimension rank(V )−a for integer a ≥ 1, so D(V (N ), 1) = D(V (N )). Even though the case a = 1 is the most natural, our results hold for all a ≥ 1.
1.1 Special case: excess intersection of hypersurfaces in projective space If X = Pr and V = O (d1 ) ⊕ · · · ⊕ O (dk ), then Theorem 1.2 specializes to Corollary 1.3, a statement about hypersurfaces failing to intersect properly in projective space. Corollary 1.3 Let d1 , . . . , dk be degrees. Then, there exists N0 such that f
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