arXiv subjects

Hideo Kojima

Publications and source records attributed to Hideo Kojima.

5 recordsLinked to original sources

arXiv2026

Curves on irrational ruled surfaces whose complements are of non-general type

Let $B$ be a curve on an irrational ruled surface $X$. We prove that the logarithmic Kodaira dimension of $X-B$ equals the Iitaka dimension of $K_X+B$ and give a rough configuration of $B$ when the logarithmic Kodaira dimension of $X - B$ is less than two. Next, we study the logarithmic multicanonical system of $X-B$ when the logarithmic Kodaira dimension of $X - B$ equals one and prove that its logarithmic $m$-canonical system gives either a $\mathbb{P}^1$-fibration or an elliptic fibration if $m \geq 12$.

math.AG↗

arXiv2024

Remarks on retracts of polynomial rings in three variables in any characteristic

Let $A$ be a retract of the polynomial ring in three variables over a field $k$. It is known that if ${\rm char}\: (k) = 0$ or ${\rm tr.deg}\:_k A \not= 2$ then $A$ is a polynomial ring. In this paper, we give some sufficient conditions for $A$ to be the polynomial ring in two variables over $k$ when ${\rm char}\: (k) > 0$ and ${\rm tr.deg}\:_k A = 2$.

math.AC↗

arXiv2024

Logarithmic multicanonical systems of smooth affine surfaces of logarithmic Kodaira dimension one

Let $S$ be a smooth affine surface of logarithmic Kodaira dimension one and let $(V,D)$ be a pair of a smooth projective surface $V$ and a simple normal crossing divisor $D$ on $V$ such that $V \setminus \operatorname{Supp} D = S$. In this paper, we consider the logarithmic multicanonical system $|m(K_V + D)|$. We prove that, for any $m \geq 8$, $|m(K_V+D)|$ gives an $\mathbb{P}^1$-fibration form $V$ onto a smooth projective curve.

math.AG↗

arXiv2019

Smooth factorial affine surfaces of logarithmic Kodaira dimension zero with trivial units

This paper considers the family $\mathscr{S}_0$ of smooth affine factorial surfaces of logarithmic Kodaira dimension 0 with trivial units over an algebraically closed field $k$. Our main result (Theorem 4.1) is that the number of isomorphism classes represented in $\mathscr{S}_0$ is at least countably infinite. This contradicts the earlier classification of Gurjar and Miyanishi [5] which asserted that $\mathscr{S}_0$ has at most two elements up to isomorphism when $k=\mathbb{C}$. Thus, the classification of surfaces in $\mathscr{S}_0$ for the field $\mathbb{C}$, long thought to have been settled, is an open problem.

math.AG↗

arXiv2018

Closed polynomials and their applications for computations of kernels of monomial derivations

In this paper, we give some results on closed polynomials and factorially closed polynomial in $n$ variables. In particular, we give a characterization of factorially closed polynomials in $n$ variables over an algebraically closed field for any characteristic. Furthermore, as an application of results on closed polynomials, we determine kernels of non-zero monomial derivations on the polynomial ring in two variables over a UFD. Finally, by using this result, for a field $k$, we determine the non-zero monomial derivations $D$ on $k[x,y]$ such that the quotient field of the kernel of $D$ is not equal to the kernel of $D$ in $k(x,y)$.

math.AG↗