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Register a new userAsymptotic syzygies and higher order embeddings
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Authors
Daniele Agostini
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We show that vanishing of asymptotic p-th syzygies implies p-very ampleness for line bundles on arbitrary projective schemes. For smooth surfaces we prove that the converse holds when p is small, by studying the Bridgeland-King-Reid-Haiman correspondence for tautological bundles on the Hilbert scheme of points. This extends previous results of Ein-Lazarsfeld, Ein-Lazarsfeld-Yang and gives a partial answer to some of their questions. As an application of our results, we show how to use syzygies to bound the irrationality of a variety.
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We introduce and study higher order Jacobian ideals, higher order and mixed Hessians, higher order polar maps, and higher order Milnor algebras associated to a reduced projective hypersurface. We relate these higher order objects to some standard graded Artinian Gorenstein algebras, and we study the corresponding Hilbert functions and Lefschetz properties.
The revised version has two additional references and a shorter proof of Proposition 5.7. This version also makes numerous small changes and has an appendix containing a proof of the degree formula for a parametrized surface.
We show that the Waring rank of the $3 times 3$ determinant, previously known to be between $14$ and $18$, is at least $15$. We use syzygies of the apolar ideal, which have not been used in this way before. Additionally, we show that the cactus rank of the $3 times 3$ permanent is at least $14$.
Let U be a basepoint free four-dimensional subpace of the space of sections of bidegree (a,b) on X = P^1 x P^1, with a and b at least 2. The sections corresponding to U determine a regular map from X to P^3. We show that there can be at most one linear syzygy on the associated bigraded ideal I_U in the bigraded ring k[s,t;u,v]. Existence of a linear syzygy, coupled with the assumption that U is basepoint free, implies the existence of an additional special pair of minimal first syzygies. Using results of Botbol, we show that these three syzygies are sufficient to determine the implicit equation of the image of X in P^3; we also show that the singular locus must contain a line.
We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids and oriented matroids. These are Stanley-Reisner ideals of complexes of independent sets, and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular realization of Stanleys formula for their Betti numbers. For unimodular matroids our resolutions are related to hyperplane arrangements on tori, and we recover the resolutions constructed by Bayer, Popescu and Sturmfels. We resolve the combinatorial problems posed in their paper by computing Mobius invariants of graphic and cographic arrangements in terms of Hermite polynomials.
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