Affine varieties/Under geometric deformation/Transcendental not affine, algebraic affine/Fact/Proof

Proof

Take ${}C\rightarrow {\mathbb {A} }_{{\mathbb {F} }_{2}}^{1}$ to be the Monsky quartic and consider the syzygy bundle

${}{\mathcal {S}}=\operatorname {Syz} {\left(x^{4},y^{4},z^{4}\right)}(6)\,$

together with the cohomology class ${}c$ determined by ${}f=y^{3}z^{3}$. This class defines an extension

$0\longrightarrow {\mathcal {S}}\longrightarrow {\mathcal {S}}'\longrightarrow {\mathcal {O}}_{C}\longrightarrow 0$

and hence ${}{\mathbb {P} }({\mathcal {S}}^{\vee })\subset {\mathbb {P} }({{\mathcal {S}}'}^{\vee })$. Then ${}{\mathbb {P} }({{\mathcal {S}}'}^{\vee })\setminus {\mathbb {P} }({\mathcal {S}}^{\vee })$ is an example with the stated properties by the previous results.

To fact