A new preprint reports exactly 28 quasi-Galois points on the Fermat quartic surface and links non-Galois examples on smooth quartic surfaces to a specific class of K3 surfaces, a named class of geometric surfaces in algebraic geometry.
The paper studies quasi-Galois points as a generalization of Galois points. Its setting is over C, the field of complex numbers.
What counts as quasi-Galois
The authors define G[P] as the group of projection-preserving birational automorphisms associated with a point P. A point is called quasi-Galois when this group has order at least 2.
For a smooth quartic surface and a point outside it, the paper gives a necessary-and-sufficient test. The point is quasi-Galois if and only if the third polar divides the first polar. In the proof, the authors reduce this divisibility statement to a vanishing condition by applying the factor theorem.
A bridge to K3 surfaces
For a non-Galois quasi-Galois point, the surface can be written in a projective normal form with the point placed at [1 : 0 : 0 : 0]. The equation contains only even powers of X, and the associated involution changes only the sign of X. This makes the surface and involution a 2-elementary K3 surface of type (8, 8, 1).
The correspondence also works in reverse within the stated setting. A general 2-elementary K3 surface of type (8, 8, 1) gives a smooth quartic surface with a quasi-Galois point through the paper's embedding construction. Together, the two directions establish 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 the specified general K3 surfaces.
That correspondence has a defined scope. It is stated for general 2-elementary K3 surfaces of type (8, 8, 1), not for every 2-elementary K3 surface, and the supplied analysis does not establish the same result over fields other than C.
The Fermat quartic is the complete count
The most concrete enumeration is for the Fermat quartic surface. The paper reports 4 Galois points and 24 additional quasi-Galois points, for 28 quasi-Galois points in all. That exact total is a special-case classification, not a number assigned to smooth quartic surfaces generally.
For every smooth quartic surface, the paper establishes that the number of quasi-Galois points is finite. It does not give one universal total for all such surfaces; the total of 28 is reported for the Fermat case.
More than one special point
When a smooth quartic surface has two distinct quasi-Galois points, the authors place it into one of two projective-equivalence normal-form families. One family separates the X,Y terms from the Z,W terms. The other uses the displayed X,W expressions together with homogeneous polynomials in Y and Z.
An explicit smooth quartic surface in the paper has exactly three quasi-Galois points: [1 : 0 : 0 : 0] and [1 : 0 : 0 : plus or minus eta], where eta squared equals -3. No two of those three points form a G-pair.
A mathematical result, not an empirical forecast
These are proof-based statements and classifications, so statistical uncertainty does not apply. The supplied review notes that some proof steps, including those used in the Fermat count, rely on cited prior results that were not independently checked in the extraction.
The document is a version 1 preprint on arXiv dated 26 August 2026, with no journal publication listed in the supplied metadata. The acknowledgment says the work was supported by JSPS KAKENHI grants JP18K03230 and JP23K03036.
Open questions identified in the analysis include whether the K3 correspondence extends beyond the stated general type (8, 8, 1) setting, whether similar criteria and exact counts hold over other fields, and how quasi-Galois points should be counted for arbitrary smooth quartic surfaces beyond the Fermat case.
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
Original paper · Full text