Proof.
Step 1: the quadruple $a$. Since $\mathcal O_ X(H)$ is globally generated, a common zero at a fixed point imposes four independent linear conditions on a quadruple $a$. The incidence variety of quadruples with a common zero at some point of the surface therefore has codimension at least two in the space of quadruples, and its image there is closed by properness. Linear independence is also a nonempty open condition, because $h^0(H)\geq 4$, and the two open conditions intersect. Fix such an $a$ and write
\[ V=H^0(X,\mathcal O_ X(3H))^{\oplus 4},\qquad N=\dim V. \]
Step 2: local freeness. At a fixed point $x$, the map $(a,b)$ drops rank exactly when $b(x)$ is proportional to the nonzero vector $a(x)$. Since evaluation on $V$ is surjective, this is a condition of codimension three. Its incidence over $X$ has dimension at most $N-1$, and its image in $V$ is closed. Outside that image the cokernel is locally free of rank two.
Step 3: the map to projective space. Dualizing the sequence identifies $E_{\mathrm{amp}}^*$ with a subbundle of $(\mathbb C^4)^*\otimes \mathcal O_ X$. This inclusion defines a proper morphism
\[ q_ b:\mathbb P(E_{\mathrm{amp}}^*)\longrightarrow \mathbb P((\mathbb C^4)^*), \qquad \mathcal O_{\mathbb P(E_{\mathrm{amp}}^*)}(1) =q_ b^*\mathcal O_{\mathbb P^3}(1). \]
For $w=[w_0:\cdots :w_3]$, its fiber is the common zero scheme of
\[ s_ w=\sum _ iw_ i a_ i\in H^0(X,\mathcal O_ X(H)), \qquad t_ w=\sum _ iw_ i b_ i\in H^0(X,\mathcal O_ X(3H)). \]
Linear independence of $a$ gives $s_ w\neq 0$ for every $w$.
Step 4: no common curves. A nonzero effective subdivisor $D$ of $\operatorname {div}(s_ w)$ has class $uS+vf$, where
\[ u\in \{ 0,1\} ,\qquad 0\leq v\leq 3e+1, \qquad (u,v)\neq (0,0). \]
Write $|D|=\mathbb P H^0(X,\mathcal O_ X(D))$ for the complete linear system. For each of these finitely many classes $D_0$, let $T_{D_0}$ parametrize the triples $(w,[\sigma ],[\tau ])$ with
\[ [\sigma ]\in |D_0|,\qquad [\tau ]\in |H-D_0|, \qquad [s_ w]=[\sigma \tau ]. \]
Since products of nonzero sections are nonzero, this is a closed projective parameter scheme. Its fiber over $w$ consists of the finitely many effective subdivisors of the fixed divisor $\operatorname {div}(s_ w)$, counted with their possible multiplicities. Thus $\dim T_{D_0}\leq 3$.
Vanishing on a nonzero effective divisor $D$ imposes at least four independent conditions on $H^0(X,\mathcal O_ X(3H))$. Indeed, choose an integral curve $C$ in its support. Very ampleness gives sections $r_0,r_1$ of $\mathcal O_ X(H)$ with nonconstant ratio on $C$. The four cubic monomials in their restrictions are linearly independent, since a linear relation would make the nonconstant ratio algebraic over $\mathbb C$, hence constant.
For fixed $(w,[\sigma ],[\tau ])$, the map $V\to H^0(X,\mathcal O_ X(3H))$, $b\mapsto t_ w$, is surjective. The condition that $t_ w$ vanish on $D=\operatorname {div}(\sigma )$ therefore has codimension at least four in $V$. This condition is closed in families: the universal multiplication map
\[ H^0(X,\mathcal O_ X(3H-D_0))\otimes \mathcal O_{|D_0|}(-1) \longrightarrow H^0(X,\mathcal O_ X(3H))\otimes \mathcal O_{|D_0|} \]
is injective on fibers, so its image is a subbundle, and the condition is the vanishing of the image of $t_ w$ in the quotient bundle. Its incidence over $T_{D_0}$ has dimension at most $N-1$, and its image in $V$ is closed.
Step 5: conclusion. Choose $b$ outside these finitely many proper closed images and outside the image of Step 2. Then $s_ w$ and $t_ w$ have no common curve for any $w$, so every fiber of $q_ b$ is finite. A proper morphism with finite fibers is finite, and the pullback of $\mathcal O_{\mathbb P^3}(1)$ by a finite morphism is ample; hence $E_{\mathrm{amp}}$ is ample. Finally, the Whitney formula gives
\[ c(E_{\mathrm{amp}})=\frac1{(1-H)(1-3H)}=1+4H+13H^2. \]
$\square$
Pome
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