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On degenerate sections of vector bundles
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We consider the locus of sections of a vector bundle on a projective scheme that vanish in higher dimension than expected. We show that after applying a high enough twist, any maximal component of this locus consists entirely of sections vanishing along a subscheme of minimal degree. In fact, we will give a more refined description of this locus, which will allow us to deduce its limit in the Grothendieck ring of varieties.
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A note on $r$ hypersurfaces intersecting in $\mathbb{P}^r$
For every choice of degrees, the largest nondegenerate component of the locus where r forms in P^r have positive-dimensional common vanishing is the family of forms all vanishing on a common line.
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