Tight closure/2 dim, standard-graded/interpretation with syzygzy bundles/description

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Let be a two-dimensional standard-graded normal domain over an algebraically closed field . Let be the corresponding smooth projective curve and let

be an -primary homogeneous ideal with generators of degrees . Then we get on the short exact sequence

Here is a vector bundle, called the syzygy bundle, of rank and of degree

An element

defines a cohomology class

With this notation we have

(homogeneous of some degree ) such that for all . This cohomology class lives in

For the plus closure we have a similar correspondence:

if and only if there exists a curve such that the pull-back of the cohomology class vanishes.