An upper jump of h0 in a flat projective family

The gluing datum. On $U_0$ let ${\mathcal{F}}_0={\mathcal{O}}_{U_0}^{\oplus 2}$ with frame $u_0,v_0$ and on $U_1$ let ${\mathcal{F}}_1={\mathcal{O}}_{U_1}^{\oplus 2}$ with frame $u_1,v_1$. On the overlap, which is $D(z)\subseteq U_0$ mapped isomorphically to $D(z^{-1})\subseteq U_1$ by $z\leftrightarrow z^{-1}$ by [F1], define ${\phi }_{10}:{\mathcal{F}}_0{\mathrm{\mid }}_{U_0\cap U_1}\to {\mathcal{F}}_1{\mathrm{\mid }}_{U_0\cap U_1}$ by $u_0\mapsto z^2{u}_1$ and $v_0\mapsto -az{u}_1+v_1$, equivalently $u_1=z^{-2}{u}_0$, $v_1=v_0+a{z}^{-1}{u}_0$. The matrix $\left(\begin{array}{cc}{z}^{-2}& a{z}^{-1}\\ 0& 1\end{array}\right)$ has determinant $z^{-2}$, a unit on the overlap, so ${\phi }_{10}$ is an isomorphism; with ${\phi }_{01}={\phi }_{10}^{-1}$ and ${\phi }_{00}={\phi }_{11}=\mathrm{id}$, the cocycle condition is vacuous on a two-element cover. By [F2] the datum glues to an ${\mathcal{O}}_X$-module $\mathcal{E}$, locally free of rank two, with $\mathcal{E}{\mathrm{\mid }}_{U_0}\cong {\mathcal{O}}_{U_0}^{\oplus 2}$ in the frame $u_0,v_0$ and similarly over $U_1$.

The sub-line-bundle and the quotient. The submodules ${\mathcal{O}}_{U_0}{u}_0$ and ${\mathcal{O}}_{U_1}{u}_1$ are identified by ${\phi }_{10}$ because $u_0\mapsto z^2{u}_1$ and $z^{-2}{u}_0=u_1$, so by [F2] they glue to a rank-one submodule $\mathcal{L}\subseteq \mathcal{E}$ with $\mathcal{L}{\mathrm{\mid }}_{U_0}=\mathcal{O}{u}_0$ and $\mathcal{L}{\mathrm{\mid }}_{U_1}=\mathcal{O}{u}_1$; its transition is $u_1=z^{-2}{u}_0$, which by [F1] is the transition of ${\mathcal{O}}_X(-2)$, so $\mathcal{L}\cong {\mathcal{O}}_X(-2)$ (both are invertible modules glued from trivialisations with the same transition function, and they agree on the overlap identifications). Likewise the classes ${\bar{v}}_i$ of $v_i$ in the quotients glue with transition ${\bar{v}}_1={\bar{v}}_0$, so the quotient $\mathcal{E}\mathrm{/}\mathcal{L}$ is isomorphic to ${\mathcal{O}}_X$. Checking on the two charts, the kernel of $\mathcal{E}\to {\mathcal{O}}_X$ is exactly $\mathcal{L}$, so $0\to {\mathcal{O}}_X(-2)\to \mathcal{E}\to {\mathcal{O}}_X\to 0$ is exact, and the extension cocycle with respect to $\{U_0,U_1\}$ is the off-diagonal entry $a{z}^{-1}$ read in the frame $u_0=x_0^{-2}$ of ${\mathcal{O}}_X(-2){\mathrm{\mid }}_{U_0}$, that is, $a{x}_0^{-1}{x}_1^{-1}$. On a fibre over $u$, lift the global section $1$ of the quotient $\mathcal{O}$ by $v_0$ and $v_1$ on the two affine charts. Their overlap difference is $v_1-v_0=a(u)z^{-1}{u}_0$; the Čech comparison and class calculation of [F5] therefore give ${\delta }_u(1)=a(u)[1\mathrm{/}(x_0{x}_1)]$ directly.

Flatness and finite presentation. By [F3] the coordinate rings $k[a,z]$, $k[a,z^{-1}]$ are free, hence flat, over $k[a]$, so the free modules $\mathcal{E}{\mathrm{\mid }}_{U_0}\cong {\mathcal{O}}_{U_0}^{\oplus 2}$ and $\mathcal{E}{\mathrm{\mid }}_{U_1}\cong {\mathcal{O}}_{U_1}^{\oplus 2}$ are flat over $S$ and finitely presented; flatness and finite presentation are local, so $\mathcal{E}$ is flat over $S$ and finitely presented, and $X\to S$ is proper of finite presentation since ${\mathbb{P}}_S^1\to S$ is projective.

Fibre sections. Fix $u\in S$ with residue field $\kappa =\kappa (u)$ and scalar $\lambda =a(u)\in \kappa$; the fibre $X_u={\mathbb{P}}_{\kappa }^1$ has the two charts $\mathrm{Spec}\kappa [z]$, $\mathrm{Spec}\kappa [z^{-1}]$, the pullback ${\mathcal{E}}_u$ has the same gluing datum with $a$ replaced by $\lambda$, and, by [F4], $h^0({\mathcal{E}}_u)=1$ when $\lambda =0$ and $h^0({\mathcal{E}}_u)=0$ when $\lambda \mathrm{\ne }0$. Indeed, a global section is given by $a_1(z)u_0+b_1(z)v_0$ on $U_0$ and $c_1(z^{-1})u_1+d_1(z^{-1})v_1$ on $U_1$ with $a_1,b_1\in \kappa [z]$, $c_1,d_1\in \kappa [z^{-1}]$; rewriting the $U_1$-expression in the frame of $U_0$ over the overlap and comparing coefficients gives $b_1=d_1$ and $a_1=z^{-2}{c}_1+\lambda z^{-1}{b}_1$. If $\lambda =0$ then $a_1=z^{-2}{c}_1$ with $a_1\in \kappa [z]$, $c_1\in \kappa [z^{-1}]$, which forces $c_1=a_1=0$, while $b_1=d_1\in \kappa [z]\cap \kappa [z^{-1}]=\kappa$ by [F1]; the solutions form the one-dimensional space spanned by the section restricting to $v_0$ and $v_1$. If $\lambda \mathrm{\ne }0$ then $b_1=d_1=\beta \in \kappa$ by [F1], and $z^2{a}_1=(\lambda \beta )z+c_1$ is an equality in $\kappa [z,z^{-1}]$. Its coefficient of $z$ gives $\lambda \beta =0$ because the left side has only degrees at least $2$ and $c_1$ has only nonpositive degrees. Then $z^2{a}_1=c_1$ lies in $z^2\kappa [z]\cap \kappa [z^{-1}]=0$, so $a_1=c_1=0$: the only global section is zero.

The jump. The origin $(a)\in S$ has residue field $k$ and $\lambda =a((a))=0$, so $h^0({\mathcal{E}}_{(a)})=1$ by 1.4 with basis the section restricting to $v_0$ and $v_1$; every other point $u\mathrm{\ne }(a)$ has $a(u)\mathrm{\ne }0$: if the corresponding prime contained $a$, it would contain the maximal ideal $(a)$ and therefore equal $(a)$; this covers closed points of arbitrary residue degree as well as the generic point — so $h^0({\mathcal{E}}_u)=0$ by 1.4. Both assertions of the statement follow, and the fibres are ${\mathbb{P}}_{\kappa (u)}^1$.

Boundaries and consistency. The field $k$ is arbitrary, including $k={\mathbb{F}}_2$; the special fibre is nonempty and ${\mathcal{E}}_{(a)}\cong \mathcal{O}(-2)\oplus \mathcal{O}$ there because the transition matrix is diagonal when $a=0$, consistent with $h^0=1$. Since $h^0$ takes the value $1$ at the origin and $0$ on the nonempty open complement $S\setminus \{(a)\}$, the function is upper semicontinuous in the sense of [F6] and the example shows that the jump is an upward one at the origin, so the conclusion of [F6] cannot be improved to local constancy of $h^0$; the long exact sequence of [F5] gives the same kernel and cokernel description of the connecting map ${\delta }_u$ (multiplication by $a(u)$), so vanishing of the class at the origin is exactly the failure of the connecting map to be injective. The Axiom of Choice and the Axiom of Dependent Choice are inherited through [F6] and the gluing and cohomology suppliers of [F2], [F4] and [F5]; no further family is selected.