Characterization of non-F-split del Pezzo surfaces of degree 2. (English) Zbl 1391.14074

Let \(k\) be an algebraically closed field of positive characteristic \(p\) and \(X\) be a scheme over \(k\). We say \(X\) is Frobenius split if the Frobenius morphism on the structure sheaf \(\mathcal{O}_{X} \to F_{X*}\mathcal{O}_{X}\) splits as a morphism of \(\mathcal{O}_{X}\)-modules. When \(X\) is a Del Pezzo surface of degree 2 and the base characteristic \(p\) is greater than 3, it is known that \(X\) is Frobenius split [N. Hara, Am. J. Math. 120, No. 5, 981–996 (1998; Zbl 0942.13006)]. In the paper under review, the author found equivalence conditions for a Del Pezzo surface of degree 2 over a field of characteristic 2 or 3 to be non Frobenius split. Assume \(X\) is a blow up of 7 points \(P_{1}, \cdots , P_{7}\) in \(\mathbb{P}^{2}\). The anticanonical system of \(X\), \(|-K_{X}|\) defines a degree 2 map, \(\pi : X \to \mathbb{P}^{2}\). The main result of this paper (Theorem 0.3) states that the followings are equivalent.
1. \(X\) is not Frobenius split.
2. The branch locus of the double cover \(\pi : X \to \mathbb{P}^{2}\) is isomorphic to the Fermat quartic (if \(p=3\)) or a double line (if \(p=2\))
3. Each conic passing through 5points in \(\{ P_{1}, \cdots , P_{7} \}\) is tangent to the line passing through remaining 2 points.
4. Each smooth member of \(|-K_{X}|\) is a supersingular elliptic curve.
5. If \(p=3\), the set \(\{P_{1}, \cdots , P_{7} \}\) is projectively equivalent to \(\{ [1,0,0], [0,1,0], [0,0,1], [1,1,1], [1,-1, \alpha] , [1,-\alpha ^{3}, \alpha ^{3}], [1,-\alpha , -1] \}\) where \(\alpha ^{2}-\alpha -1 =0\).


14J26 Rational and ruled surfaces


Zbl 0942.13006
Full Text: DOI Euclid