Seshadri constants for vector bundles

On the Seshadri constants of equivariant bundles over Bott-Samelson varieties and wonderful compactifications

“…where the infimum is taken over all curves C ⊂ P(E) that intersect p −1 (x), but are not contained in p −1 (x). For more details on Seshadri constants of vector bundles, see [14,13].…”

“…Remark 3.7. It is known that a nef vector bundle E on a projective variety X is ample if and only if inf x∈X ε(E, x) > 0; for example, see [13,Theorem 3.11]. On the other hand, in general, Seshadri constants of ample vector bundles can be arbitrarily small positive numbers (see [21,Example 5.2.1]).…”

“…Theorem 3.5 shows that for an ample T -equivariant vector bundle E on Z(w, i), we have ε(E, x) ≥ 1 for every T -fixed point x ∈ Z(w, i). If E is a vector bundle on a projective variety X, then the maximum value of ε(E, x), as the point x runs over X, is achieved at a very general point in X (see [13,Proposition 3.35]). Since T -fixed points are special, we conclude that ε(E, x) ≥ 1 for a very general point x ∈ Z(w, i) if E is an ample T -equivariant vector bundle on Z(w, i).…”

“…An explicit definition of these was first given by Hacon [14]; see Definition 3.4 below. For a comprehensive survey and a generalization to the relative setting, see [13].…”

Seshadri constants of equivariant vector bundles on toric varieties

“…Seshadri constants for ample vector bundles on smooth projective curves are computed in [H] and for nef vector bundles in [FM]. The Seshadri constant of a discriminant zero semistable vector bundle can be computed from the determinant line bundle (see [FM,Lemma 3.27]). Seshadri constants for direct sum of vector bundles are the minimum of Seshadri constants of the component bundles (see [FM,Lemma 3.31]).…”