The non-symplectic index of supersingular K3 surfaces
In this paper, we found non-symplectic index of all supersingular K3 surfaces defined over a field of characteristic p>3.
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arXiv subjects
Junmyeong Jang
Publications and source records attributed to Junmyeong Jang.
A non-symplectic automorphism of order 21 of a K3 surface
In this paper, we prove that, over an algebraically closed field whose characteristic is not 2,3 nor 7, a pair of a K3 surface and a purely non-symplectic automorphism of order 21 or 42 is unique up to isomorphism.
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The representations of the automorphism groups and the Frobenius invariants of K3 surfaces
For a K3 surface over an algebraically closed field of odd characteristic, the representation of the automorphism group on the global two forms is finite. If the K3 surface is supersingular, it is isomorphic to the representation on the discriminant group of the Neron-Severi group. If the K3 surface is of finite height, the representation on the (etale or crystalline) transcendental cycles is also finite and there is a canonical projection from the representation on the transcendental cycles to the representation on the two forms. We prove that, if the base field is an algebraic closure of a finite field, this projection is an isomorphism and the rank of the transcendental cycles is a multiple of the order of the representation on the two forms. They are analogous to the classical results on complex K3 surfaces. From these results, we deduce that the height and the Artin invariant of a K3 surface with a purely non-symplectic automorphism of higher order are determined by a congruence class of the base characteristic.
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Neron-Severi group preserving lifting of K3 surfaces and applications
For a K3 surface of finite height over a field of odd characteristic, there exists a smooth lifting to the ring of Witt vectors such that the reduction map from the Picard group of the generic fiber to the Picard group of the special fiber is isomorphic. In this paper, using this result, we give a criterion for a K3 surface of finite height over a field of odd characteristic to be an Enriques K3 surface or to be a K3 surface in terms of the Neron-Severi lattice. Then we show every Kummer surface has an Enriques involution. We also give a classification of K3 surfaces of Picard rank of 20 over odd characteristic.
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A lifting of an automorphism of a K3 surface over odd characteristic
In this paper, we prove that, over an algebraically closed field of odd characteristic, a weakly tame automorphism of a K3 surface of finite height can be lifted over the ring of Witt vectors of the base field. Also we prove that a non-symplectic tame automorphism of a supersingular K3 surface or a symplectic tame automorphism of a supersingular K3 surface of Artin-invariant at least 2 can be lifted over the ring of Witt vectors. Using these results, we prove, for a weakly tame K3 surface of finite height, there is a lifting over the ring of Witt vectors to which whole the automorphism group of the K3 surface can be lifted. Also we prove a K3 surface equipped with a purely non-symplectic automorphism of a certain order is unique up to isomorphism.
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An Enriques involution of a supersingular K3 surface over odd characteristic
In this paper, we prove, as the complex case, a supersingular K3 surface over a field of odd characteristic has an Enriques involution if and only if there exists a primitive embedding of the twice of the Enriques lattice into the Neron-Severi group such that the orthogonal complement of the embedding has no vector of self-intersection -2 using the Crystalline Torelli theorem. By this criterion and some lattice calculation, we prove that when the characteristic of the base field is p=19 or p>23, a superingular K3 surface is an Enriques K3 surface if and only if the Artin invariant is less than 6.
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The ordinarity of an insotrivial elliptic fibration
In this paper, we study the ordinarity of an isotrivial elliptic surface defined over a field of positive characteristic. If an isotrivial elliptic fibration $π:X \to C$ is given, $X$ is ordinary when the common fiber of $π$ is ordinary and a certain finite cover of the base $C$ is ordinary. By this result, we may obtain the ordinary reduction theorem for some kinds of isotrivial elliptic fibrations.
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Generic ordinarity for semi-stable fibration
In this paper, we proved that, for a semi-stable fibration of a proper smooth surface to a proper smooth curve over a field of positive characteristic, if the generic fiber is ordinary, then the semi-positivity theorem holds. As an application, we constrcuted a counterexample to Parshin's conjecture on the Miyaoka-Yau inequality.
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Semi-stable fibration of generic p-rank 0
In this paper, we proved that, for a semi-stable fibration of a proper smooth surface to a proper smooth curve over a filed of positive characteristic, if the p-rank of the generic fiber is 0, then the base change of the fibration by a sufficiently many iterative Frobenius morphism of the base curve violates the semi-positivity theorem. As an application, we suggest a statement on a distribution of p-ranks of reductions for a certain non-closed point in the moduli space over a number field.
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