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Ichiro Shimada

Publications and source records attributed to Ichiro Shimada.

At least 55 records · Page 3Linked to original sources
arXiv2009

Lattice Zariski k-ples of plane sextic curves and Z-splitting curves for double plane sextics

A simple sextic is a reduced complex projective plane curve of degree 6 with only simple singularities. We introduce a notion of Z-splitting curves for the double covering of the projective plane branching along a simple sextic, and investigate lattice Zariski k-ples of simple sextics by Z-splitting curves. We define specialization of lattice types, and classify all lattice types of Z-splitting curves of degree less than or equal to 3 up to specializations.

math.AG↗
arXiv2007

Non-homeomorphic conjugate complex varieties

We present a method to produce examples of non-homeomorphic conjugate complex varieties based on the genus theory of lattices. As an application, we give examples of arithmetic Zariski pairs.

math.AG↗
arXiv2007

Transcendental lattices and supersingular reduction lattices of a singular $K3$ surface

A (smooth) K3 surface X defined over a field k of characteristic 0 is called singular if the Néron-Severi lattice NS (X) of X over the algebraic closure of k is of rank 20. Let X be a singular K3 surface defined over a number field F. For each embedding σof F into the complex number field, we denote by T(X^σ) the transcendental lattice of the complex K3 surface X^σobtained from X by σ. For each prime ideal P of F at which X has a supersingular reduction X_P, we define L(X, P) to be the orthogonal complement of NS(X) in NS(X_P). We investigate the relation between these lattices T(X^σ) and L(X, P). As an application, we give a lower bound of the degree of a number field over which a singular K3 surface with a given transcendental lattice can be defined.

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arXiv2006

On normal K3 surfaces

We determine all possible configurations of rational double points on complex normal algebraic K3 surfaces, and on normal supersingular K3 surfaces in characteristic p > 19.

math.AG↗
arXiv2006

On elliptic K3 surfaces

We classify all the possible configurations of singular fibers and the torsion parts of Mordell-Weil groups of complex elliptic K3 surfaces. The complete list of 3279 configurations is attached.

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arXiv2004

K3 surfaces with ten cusps

We show that normal K3 surfaces with ten cusps exist in and only in characteristic 3. We determine these K3 surfaces according to the degrees of the polarizations. Explicit examples are given.

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arXiv2003

Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases

Let $X$ be a general complete intersection of a given multi-degree in a complex projective space. Suppose that the anti-canonical line bundle of $X$ is ample. Using the cylinder homomorphism associated with the family of complete intersections contained in $X$, we prove that the vanishing cycles in the middle homology group of $X$ are represented by topological cycles whose support is contained in a proper Zariski closed subset $T\subset X$ of certain codimension. In some cases, we can find such a Zariski closed subset $T$ with codimension equal to the upper bound obtained from the Hodge structure of the middle cohomology group of $X$ by means of Gröbner bases. Hence a consequence of the generalized Hodge conjecture is verified in these cases.

math.AG↗
arXiv2003

Supersingular K3 surfaces in characteristic 2 as double covers of a projective plane

For every supersingular $K3$ surface $X$ in characteristic 2, there exists a homogeneous polynomial $G$ of degree 6 such that $X$ is birational to the purely inseparable double cover of a projective plane defined by $w^2=G$. We present an algorithm to calculate from $G$ a set of generators of the numerical Néron-Severi lattice of $X$. As an application, we investigate the stratification defined by the Artin invariant on a moduli space of supersingular $K3$ surfaces of degree 2 in characteristic 2.

math.AG↗
arXiv2003

Rational double points on supersingular K3 surfaces

We investigate configurations of rational double points with the total Milnor number 21 on supersingular $K3$ surfaces. The complete list of possible configurations is given. As an application, we also give the complete list of extremal (quasi-)elliptic fibrations on supersingular $K3$ surfaces.

math.AG↗
arXiv1995

A note on Zariski pairs

A couple of complex projective plane curves are said to make a Zariski pair if they have the same degree and the same type of singularities, but their embeddings in the projective plane are topologically different. In this paper, we present a method to construct certain type of Zariski pairs, and make infinite series of examples.

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