Remark 107.5.2. The boundedness argument in the proof of Lemma 107.5.1 does not work for moduli of surfaces and in fact, the result is wrong, for example because K3 surfaces over fields can have infinite discrete automorphism groups. The “reason” the argument does not work is that on a projective surface $S$ over a field, given ample invertible sheaves $\mathcal{N}$ and $\mathcal{L}$ with Hilbert polynomials $Q$ and $P$, there is no a priori bound on the Hilbert polynomial of $\mathcal{N} \otimes _{\mathcal{O}_ S} \mathcal{L}$. In terms of intersection theory, if $H_1$, $H_2$ are ample effective Cartier divisors on $S$, then there is no (upper) bound on the intersection number $H_1 \cdot H_2$ in terms of $H_1 \cdot H_1$ and $H_2 \cdot H_2$.

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