## 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 $$\alpha ^{2}-\alpha -1 =0$$.

### MSC:

 14J26 Rational and ruled surfaces

### Keywords:

Del Pezzo surface of degree 2; Frobenius split

Zbl 0942.13006
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