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

Let ${}R$ be a two-dimensional standard-graded normal domain over an algebraically closed field ${}K$. Let ${}C=\operatorname {Proj} {\left(R\right)}$ be the corresponding smooth projective curve and let

${}I={\left(f_{1},\ldots ,f_{n}\right)}\,$

be an ${}R_{+}$-primary homogeneous ideal with generators of degrees ${}d_{1},\ldots ,d_{n}$. Then we get on ${}C$ the short exact sequence

$0\longrightarrow \operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m)\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(m-d_{i}){\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}{\mathcal {O}}_{C}(m)\longrightarrow 0.$

Here ${}\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m)$ is a vector bundle, called the syzygy bundle, of rank ${}n-1$ and of degree

$((n-1)m-\sum _{i=1}^{n}d_{i})\operatorname {deg} \,(C).$

An element

${}f\in R_{m}=\Gamma (C,{\mathcal {O}}_{C}(m))\,$

defines a cohomology class

${}c=\delta (f)\in H^{1}(C,\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m))\,.$

With this notation we have

$f\in I^{*}{\text{ if and only if there exists }}z\neq 0$

(homogeneous of some degree ${}k$) such that ${}zF^{e*}(c)=0$ for all ${}e$. This cohomology class lives in

${}H^{1}(C,F^{e*}(\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m))\otimes {\mathcal {O}}(k))=H^{1}(C,\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(qm+k))\,.$

For the plus closure we have a similar correspondence:

${}f\in I^{+}$ if and only if there exists a curve ${}\varphi \colon C'\rightarrow C$ such that the pull-back ${}\varphi ^{*}(c)$ of the cohomology class vanishes.