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

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

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

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

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

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

Recall that the degree of a vector bundle ${\displaystyle {}{\mathcal {S}}}$ on a projective curve is defined as the degree of the invertible sheaf ${\displaystyle {}\bigwedge ^{r}{\mathcal {S}}}$, where ${\displaystyle {}r}$ is the rank of ${\displaystyle {}{\mathcal {S}}}$. The degree is additive on short exact sequences.

A homogeneous element ${\displaystyle {}f}$ of degree ${\displaystyle {}m}$ defines an element in ${\displaystyle {}\Gamma (C,{\mathcal {O}}_{C}(m))}$ and thus a cohomology class ${\displaystyle {}\delta (f)\in H^{1}(C,\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m))}$, so this defines a torsor over the projective curve. We mention an alternative description of the torsor corresponding to a first cohomology class in a locally free sheaf which is better suited for the projective situation.

## Remark

Let ${\displaystyle {}{\mathcal {S}}}$ denote a locally free sheaf on a scheme ${\displaystyle {}X}$. For a cohomology class ${\displaystyle {}c\in H^{1}(X,{\mathcal {S}})}$ one can construct a geometric object: Because of ${\displaystyle {}H^{1}(X,{\mathcal {S}})\cong \operatorname {Ext} ^{1}({\mathcal {O}}_{X},{\mathcal {S}})}$, the class defines an extension

${\displaystyle 0\longrightarrow {\mathcal {S}}\longrightarrow {{\mathcal {S}}'}\longrightarrow {\mathcal {O}}_{X}\longrightarrow 0.}$

This extension is such that under the connecting homomorphism of cohomology, ${\displaystyle {}1\in \Gamma (X,{\mathcal {O}}_{X})}$ is sent to ${\displaystyle {}c\in H^{1}(X,{\mathcal {S}})}$. The extension yields a projective subbundle[1]

${\displaystyle {}{\mathbb {P} }({\mathcal {S}}^{\vee })\subset {\mathbb {P} }({{\mathcal {S}}'}^{\vee })\,.}$

If ${\displaystyle {}V}$ is the corresponding geometric vector bundle of ${\displaystyle {}{\mathcal {S}}}$, one may think of ${\displaystyle {}{\mathbb {P} }({\mathcal {S}}^{\vee })}$ as ${\displaystyle {}{\mathbb {P} }(V)}$ which consists for every base point ${\displaystyle {}x\in X}$ of all the lines in the fiber ${\displaystyle {}V_{x}}$ passing through the origin. The projective subbundle ${\displaystyle {}{\mathbb {P} }(V)}$ has codimension one inside ${\displaystyle {}{\mathbb {P} }(V')}$, for every point it is a projective space lying (linearly) inside a projective space of one dimension higher. The complement is then over every point an affine space. One can show that the global complement

${\displaystyle {}T={\mathbb {P} }({{\mathcal {S}}'}^{\vee })\setminus {\mathbb {P} }({\mathcal {S}}^{\vee })\,}$

is another model for the torsor given by the cohomology class. The advantage of this viewpoint is that we may work, in particular when ${\displaystyle {}X}$ is projective, in an entirely projective setting.

1. ${\displaystyle {}{\mathcal {S}}^{\vee }}$ denotes the dual bundle. According to our convention, the geometric vector bundle corresponding to a locally free sheaf ${\displaystyle {}{\mathcal {T}}}$ is given by ${\displaystyle {}\operatorname {Spec} {\left(\oplus _{k\geq 0}S^{k}({\mathcal {T}})\right)}}$ and the projective bundle is ${\displaystyle {}\operatorname {Proj} {\left(\oplus _{k\geq 0}S^{k}({\mathcal {T}})\right)}}$, where ${\displaystyle {}S^{k}}$ denotes the ${\displaystyle {}k}$th symmetric power.