The Fermat quartic has exactly 28 quasi-Galois points. The count separates into four Galois points and 24 quasi-Galois points that are not Galois. The 28-point figure is specific to the Fermat quartic. The paper’s broader result is that every smooth quartic surface has finitely many quasi-Galois points.
A point is called quasi-Galois when the group of birational automorphisms preserving the projection from that point has at least two elements. In other words, at least two such symmetries preserve the projection. The analysis concerns smooth quartic surfaces in three-dimensional projective space.
A test based on divisibility
For points outside the surface, the paper gives an if-and-only-if test: a point is quasi-Galois exactly when its third polar is divisible by its first polar. The criterion turns the question into a divisibility check, but its stated scope stops at points outside the surface.
In the non-Galois case, projective equivalence puts the point at [1 : 0 : 0 : 0]. A generating symmetry then changes the sign of X while leaving Y, Z and W unchanged. The paper uses this sign change to describe the surface in an even-in-X quartic normal form.
A bridge to K3 surfaces
The paper also connects these points with a class of objects known as 2-elementary K3 surfaces. Its framework classifies a surface paired with its relevant symmetry by a type written as (r, a, delta). It then states a one-to-one correspondence between projective-equivalence classes of pairs consisting of a smooth quartic surface and a non-Galois quasi-Galois point, and isomorphism classes of general 2-elementary K3 surfaces of type (8, 8, 1), through a quasi-Galois embedding.
In the converse construction, a degree-two Hurwitz calculation leaves the relevant intersection number m with two possible values, 0 or 4. The paper concludes from this calculation that the associated map is an embedding, completing the route from the K3 description back to the quartic setting.
When several points are present
Beyond the one-point correspondence, the paper gives a classification result for smooth quartic surfaces with two distinct quasi-Galois points. Up to projective equivalence, those surfaces fall into two normal-form branches.
The paper also gives a smooth quartic example with exactly three quasi-Galois points. No two of those three points form a G-pair.
Why 28 is a special-case count
For the Fermat quartic, the paper organizes the relevant points into coordinate-pair families indexed by i equal to 0, 1, 2 and 3. The reported breakdown is four Galois points and 24 non-Galois quasi-Galois points, for 28 quasi-Galois points altogether. The result is exact within the stated theorem conditions.
There are clear boundaries to the conclusions. The K3 correspondence is stated for non-Galois quasi-Galois points and general 2-elementary K3 surfaces of type (8, 8, 1), while the polar divisibility test applies to points outside the surface. Neither result, on its own, supplies a 28-point count for an arbitrary smooth quartic.
The supplied manuscript is an arXiv version 2 document dated 8 October 2026. The authors report support from JSPS KAKENHI Grant Numbers JP18K03230 and JP23K03036.
Paper data and sources
Original title: Quasi-Galois points for quartic surfaces
Authors: Kei Miura, Shingo Taki
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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