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Kimiko Yamada

Publications and source records attributed to Kimiko Yamada.

8 recordsLinked to original sources
arXiv2024

Perspective and open problems on birational properties and singularities of moduli scheme of sheaves on surfaces

For complex projective smooth surface $X$, let $M$ be the coarse moduli scheme of rank-two stable sheaves with fixed Chern classes. Grasping the birational structure of $M$, for example its Kodaira dimension, is a fundamental problem. However, in the case where $κ(X)>0$, the study of this problem has not necessarily been active in recent years. In this article we survey the study of this problem, especially for the case where $κ(X)=1$ and $c_1=0$. We will also survey some research on the structure of singularities of $M$, and a minimal model program of $M$. While explaining motivations, we raise several unsolved problems.

math.AG↗
arXiv2021

The Kodaira dimension and singularities of moduli of stable sheaves on some elliptic surfaces

Let $X$ be an elliptic surface over ${\bf P}^1$ with $κ(X)=1$, and $M=M(c_2)$ be the moduli scheme of rank-two stable sheaves $E$ on $X$ with $(c_1(E),c_2(E))=(0,c_2)$ in $\operatorname{Pic}(X)\times\mathbb{Z}$. We look into defining equations of $M$ at its singularity $E$, partly because if $M$ admits only canonical singularities, then the Kodaira dimension $κ(M)$ can be calculated. We show the following. (A) $E$ is at worst canonical singularity of $M$ if the restriction of $E_η$ to the generic fiber of $X$ has no rank-one subsheaf, and if the number of multiple fibers of $X$ is a few. (B) We obtain that $κ(M)=\{1+\dim(M)\}/2$ and the Iitaka program of $M$ can be described in purely moduli-theoretic way for $c_2\gg 0$, when $χ({\mathcal O}_X)=1$, $X$ has just two multiple fibers, and one of its multiplicities equals $2$. (C) On the other hand, when $E_η$ has a rank-one subsheaf, it may be insufficient to look at only the degree-two part of defining equations to judge whether $E$ is at worst canonical singularity or not.

math.AG↗
arXiv2008

Flips and variation of moduli schemes of sheaves on a surface

Let $H$ be an ample line bundle on a non-singular projective surface $X$, and $M(H)$ the coarse moduli scheme of rank-two $H$-semistable sheaves with fixed Chern classes on $X$. We show that if $H$ changes and passes through walls to get closer to $K_X$, then $M(H)$ undergoes natural flips with respect to canonical divisors. When $X$ is minimal and its Kodaira dimension is positive, this sequence of flips terminates in $M(H_X)$; $H_X$ is an ample line bundle lying so closely to $K_X$ that the canonical divisor of $M(H_X)$ is nef. Remark that so-called Thaddeus-type flips somewhat differ from flips with respect to canonical divisors.

math.AG↗
arXiv2008

Desingularization of some moduli scheme of stable sheaves on a surface

Let $X$ be a nonsingular projective surface over an algebraically closed field with characteristic zero, and $H_-$ and $H_+$ ample line bundles on $X$ separated by only one wall of type $(c_1,c_2)$. Suppose the moduli scheme $M(H_-)$ of rank-two $H_-$-stable sheaves with Chern classes $(c_1,c_2)$ is non-singular. We shall construct a desingularization of $M(H_+)$ by using $M(H_-)$. As an application, we consider whether singularities of $M(H_+)$ are terminal or not when $X$ is ruled or elliptic.

math.AG↗
arXiv2007

A sequence of blowing-ups connecting moduli of sheaves and the Donaldson polynomial under change of polarization

Let $H$ and $H'$ be two ample line bundles over a nonsingular projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r=2,c_1,c_2)$. In a moduli-theoretic way that comes from elementary transforms, we connect $M(H)$ and $M(H')$ by a sequence of blowing-ups when walls separating $H$ and $H'$ are not necessarily good. As an application, we also consider the polarization change problem of Donaldson polynomials.

math.AG↗
arXiv2007

Blowing-ups describing the polarization change of moduli schemes of semistable sheaves of general rank

Let $H$ and $H'$ be two ample line bundles over a smooth projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r,c_1,c_2)$. We construct a sequence of blowing-ups which describes how $M(H)$ differs from $M(H')$ not only when $r=2$ but also when $r$ is arbitrary. Means we here utilize are elementary transforms and the notion of a sheaf with flag.

math.AG↗