Homogeneous prime ideals of height h have at most h² quadratic minimal generators, and this bound is sharp as some such primes are minimally generated by exactly h² quadrics.
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Uniform effective bounds on projective dimension and Castelnuovo-Mumford regularity of homogeneous ideals, independent of number of variables and depending on a fraction of the syzygies.
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Quadratic linear strands of prime ideals
Homogeneous prime ideals of height h have at most h² quadratic minimal generators, and this bound is sharp as some such primes are minimally generated by exactly h² quadrics.
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Uniform bounds on projective dimension and Castelnuovo-Mumford regularity
Uniform effective bounds on projective dimension and Castelnuovo-Mumford regularity of homogeneous ideals, independent of number of variables and depending on a fraction of the syzygies.