Relative Seshadri constants.
The Seshadri constant of a nef divisor at a point of a projective variety is a measure of local positivity of the divisor around the point.
We introduce a relative version of this and observe how it parallels the classical case of divisors. A particular case, the Seshadri constants of vector bundles, was studied by C. Hacon in his thesis.
From the relative perspective, we are able to translate a Nagata-type conjecture on the nef cone of CxC where C is a very general curve of large genus into a question of semistability for higher syzygy bundles on C.
We also ask whether P^n is the only n-dimensional smooth projective variety whose tangent bundle has positive Seshadri constant at some point, and answer yes in several cases.
This is joint work with T. Murayama.