By singular K3 I mean a smooth complex K3 with Néron-Severi rank equal to 20.
Are singular K3 surfaces dense in the moduli space of polarized K3 surfaces?
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Sign up to join this communityBy singular K3 I mean a smooth complex K3 with Néron-Severi rank equal to 20.
Are singular K3 surfaces dense in the moduli space of polarized K3 surfaces?
This is a standard argument and there probably exists a reference but it's not hard once you rephrase it in terms of the period domain.
The moduli space of K3 surfaces is locally isomorphic to its period domain.
The period domain is an open subset of the vanishing locus of a quadratic polynomial in $\mathbb P^{21}(\mathbb C)$. See, for instance, Huybrechts' lectures on K3 surfaces.
So it suffices to show the singular points are dense in the period domain.
The singular points in the period domain correspond to points whose coordinates generate a lattice of rank two in the complex numbers. Examples include points defined over an imaginary quadratic field. (One can check that these are the only examples but that is not necessary for this argument.)
So it suffices to check that any tuple of complex numbers satisfying this quadratic polynomial can be approximated by a tuple of numbers in $\mathbb Q(i)$ satisfying this quadratic polynomial. To do this, express the polynomial as $xy+$ a bunch of terms not involving $x$ and $y$, approximate each complex number by an element of $\mathbb Q(i)$, then adjust $x$ or $y$ by a little bit to make the quadratic polynomial vanish again.
A reference for the density of singular K3 surfaces in the period domain (see Will Sawin's answer) is
Piatetski-Shapiro, Ilya I.; Shafarevich, I. R., Arithmetic of K3 surfaces, Trudy Mezhdunarod. Konf. Teor. Chisel, Moskva 1971, Tr. Mat. Inst. Steklova 132, 44-54 (1973) ZBL0293.14010.
In particular, this is a screenshot of page 54: