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10
pcyclic actions on K3 surfaces
 J. Algebraic Geometry
"... In this paper we study automorphisms g of order p of K3surfaces defined over an algebraically closed field of characteristic p> 0. We divide all possible actions in the following cases according to the structure of the set of fixed points X g: X g is a finite set, X g contains a onedimensional ..."
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Cited by 9 (4 self)
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In this paper we study automorphisms g of order p of K3surfaces defined over an algebraically closed field of characteristic p> 0. We divide all possible actions in the following cases according to the structure of the set of fixed points X g: X g is a finite set, X g contains a onedimensional part D which is a positive divisor of Kodaira dimension κ(X,D) = 0,1,2. In the latter case we
Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic
 math.AG/0403478. ICHIRO SHIMADA AND DEQI ZHANG
"... Abstract. We show that Mukai’s classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic p under the assumptions that (i) the order of the group is coprime to p and (ii) either the surface or its quotient is not birationally isomorphic to ..."
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Cited by 8 (1 self)
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Abstract. We show that Mukai’s classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic p under the assumptions that (i) the order of the group is coprime to p and (ii) either the surface or its quotient is not birationally isomorphic to a supersingular K3 surface with Artin invariant 1. In the case without the assumption (ii) we classify all possible new groups which may appear. We prove that the assumption (i) on the order of the group is always satisfied if p> 11 and if p = 2, 3, 5,11 we give examples of K3 surfaces with finite symplectic automorphism groups of order divisible by p which are not contained in Mukai’s list. 1.
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
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Cited by 8 (7 self)
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Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
THE HESSE PENCIL OF PLANE CUBIC CURVES
"... Abstract. This is a survey of the classical geometry of the Hesse configuration of 12 lines in the projective plane related to inflection points of a plane cubic curve. We also study two K3 surfaces with Picard number 20 which arise naturally in connection with the configuration. 1. ..."
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Cited by 4 (0 self)
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Abstract. This is a survey of the classical geometry of the Hesse configuration of 12 lines in the projective plane related to inflection points of a plane cubic curve. We also study two K3 surfaces with Picard number 20 which arise naturally in connection with the configuration. 1.
On exponential sums
, 1997
"... (0.1) Let k denote a finite field with q = ps elements, let f ∈ k[x1,...,xn] be a polynomial and let Ψ: Fp → C ∗ be a nontrivial additive character. Consider the exponential sum: ..."
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(0.1) Let k denote a finite field with q = ps elements, let f ∈ k[x1,...,xn] be a polynomial and let Ψ: Fp → C ∗ be a nontrivial additive character. Consider the exponential sum:
Kummer surfaces for the selfproduct of the . . .
, 2005
"... The classical Kummer construction attaches to an abelian surface a K3 surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by the selfproduct of the rational cuspidal curve, and the s ..."
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The classical Kummer construction attaches to an abelian surface a K3 surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by the selfproduct of the rational cuspidal curve, and the sign involution by suitable infinitesimal group scheme actions, I give the correct Kummertype construction for this situation. We encounter rational double points of type D4 and D8, instead of type A1. It turns out that the resulting surfaces are supersingular K3 surfaces with Artin invariant one and two. They lie in a 1dimensional family obtained by simultaneous resolution, which exists after purely inseparable base change.
Kummer surfaces for the . . .
, 2008
"... The classical Kummer construction attaches to an abelian surface a K3 surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by the selfproduct of the rational cuspidal curve, and the si ..."
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The classical Kummer construction attaches to an abelian surface a K3 surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by the selfproduct of the rational cuspidal curve, and the sign involution by suitable infinitesimal group scheme actions, I give the correct Kummertype construction in this situation. We encounter rational double points of type D4 and D8, instead of type A1. It turns out that the resulting surfaces are supersingular K3 surfaces with Artin invariant one and two. They lie in a 1dimensional family obtained by simultaneous resolution after purely inseparable base change.
Singularities to appearing on geometric generic fibers of morphism between smooth schemes
, 2006
"... I give various criteria for singularities to appear on geometric generic fibers of morphism between smooth schemes in positive characteristics. This involves local fundamental groups, jacobian ideals, projective dimension, tangent and cotangent sheaves, and the effect of Frobenius. As an application ..."
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I give various criteria for singularities to appear on geometric generic fibers of morphism between smooth schemes in positive characteristics. This involves local fundamental groups, jacobian ideals, projective dimension, tangent and cotangent sheaves, and the effect of Frobenius. As an application, I determine which rational double points do appear on geometric generic fibers.
ON KUMMER TYPE CONSTRUCTION OF SUPERSINGULAR K3 SURFACES IN CHARACTERISTIC 2
"... Abstract. We show that every supersingular K3 surface in characteristic 2 with Artin invariant ≤ 2 is obtained by the Kummer type construction of Schröer. 1. ..."
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Abstract. We show that every supersingular K3 surface in characteristic 2 with Artin invariant ≤ 2 is obtained by the Kummer type construction of Schröer. 1.
SINGULARITIES OF DUAL VARIETIES IN CHARACTERISTIC 2
, 2006
"... Abstract. We investigate unibranched singularities of dual varieties of evendimensional smooth projective varieties in characteristic 2. 1. ..."
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Abstract. We investigate unibranched singularities of dual varieties of evendimensional smooth projective varieties in characteristic 2. 1.