arXiv subjects

Sachiko Saito

Publications and source records attributed to Sachiko Saito.

7 recordsLinked to original sources

arXiv2024

Toric resolutions of strongly mixed weighted homogeneous polynomial germs of type $J_{10}^-$

We consider toric resolutions of some strongly mixed weighted homogeneous polynomials of type $J_{10}^-$. We show that the strongly mixed weighted homogeneous polynomial $f := f_{2,2,1,2,1,4}\ (k=3)$ (see §3) has no mixed critical points on ${\mathbb{C}^*}^2$ (Lemma 14), and moreover, show that the strict transform $\tilde V$ of the mixed hypersurface singularity $V := f^{-1}(0)$ via the toric modification $\hatπ : X \to \mathbb{C}^2$, where we set $f := f_{2,2,1,2,1,4}\ (k=3)$, is not only a real analytic manifold outside of ${\tilde V} \cap \hatπ^{-1}(\boldsymbol{0})$ but also a real analytic manifold as a germ of ${\tilde V} \cap \hatπ^{-1}(\boldsymbol{0})$ (Theorem 15).

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arXiv2022

A note on Newton non-degeneracy of mixed weighted homogeneous polynomials

A mixed polynomial $f(\boldsymbol{z}, \bar{\boldsymbol{z}})$ is called a mixed weighted homogeneous polynomial (Definition 5) if it is both radially and polar weighted homogeneous. Let $f$ be a mixed weighted homogeneous polynomial with respect to a strictly positive radial weight vector $P$ and a polar weight vector $Q$. Suppose that $f$ is Newton non-degenerate over a compact face $Δ(P)$ and polar weighted homogeneous of non-zero polar degree with respect to $Q$. Then $f : {\mathbb{C}^*}^n \to \mathbb{C}$ has no mixed critical points. Moreover, under the assumption $f^{-1}(0) \cap {\mathbb{C}^*}^n \neq \emptyset$, $f : {\mathbb{C}^*}^n \to \mathbb{C}$ is surjective. In other words, in this situation, Newton non-degeneracy over a compact face $Δ(P)$ implies strong Newton non-degeneracy over $Δ(P)$ (Proposition 10). With this fact as a starting point, we investigate the sets $f^{-1}(0) \cap {\mathbb{C}^*}^n$, and show the existence of a collection of mixed weighted homogeneous polynomials $f = f_{Δ(P)}$ of non-zero polar degree which satisfy $\dim Δ(P) \geq 1$ and $f^{-1}(0) \cap {\mathbb{C}^*}^n = \emptyset$ (Theorem 11). We also give an example of convenient mixed function germs of mixed weighted homogeneous face type which are not true non-degenerate (Definition 14).

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arXiv2021

Resolutions of Newton non-degenerate mixed polynomials of strongly polar non-negative mixed weighted homogeneous face type

Let $f(\mathbb{z},\bar{\mathbb{z}})$ be a convenient Newton non-degenerate mixed polynomial with strongly polar non-negative mixed weighted homogeneous face functions. We consider a convenient regular simplicial cone subdivision $Σ^*$ which is admissible for $f$ and take the toric modification $\hatπ : X \to \mathbb{C}^n$ associated with $Σ^*$. We show that the toric modification resolves topologically the singularity of the mixed hypersurface germ defined by $f(\mathbb{z},\bar{\mathbb{z}})$ under the Assumption (*) (Theorem 32). This result is an extension of the first part of Theorem 11 ([4]) by Mutsuo Oka. We also consider some typical examples (§9).

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arXiv2020

On real anti-bicanonical curves with one double point on the $4$-th real Hirzebruch surface. II

A real 2-elementary K3 surfaces of type ((3,1,1),- id) yields a real anti-bicanonical curve s \cup A^\prime_1 (disjoint union) on the 4-th real Hirzebruch surface F_4 where s is the exceptional section of F_4 and the real curve A^\prime_1 has one real double point. We give a criterion (see Proposition 2.4) which determines whether the real double point is degenerate or not. One direction of the assertion of this proposition has already been proved in Lemma 4.6 of the preceding paper [9] (2015) by the author. In this paper we prove the inverse direction.

math.AG↗

arXiv2014

On real anti-bicanonical curves with one double point on the 4-th real Hirzebruch surface

We list up all the candidates for the real isotopy types of real anti-bicanonical curves with one real nondegenerate double point on the 4-th real Hirzebruch surface RF_4 by enumerating the connected components of the moduli space of real 2-elementary K3 surfaces of type (S,θ)=((3,1,1), -id). We also list up all the candidates for the non-increasing simplest degenerations of real nonsingular anti-bicanonical curves on RF_4. We find an interesting correspondence between the real isotopy types of real anti-bicanonical curves with one real nondegenerate double point on RF_4 and the non-increasing simplest degenerations of real nonsingular anti-bicanonical curves on RF_4. This correspondence is very similar to the one provided by the rigid isotopic classification of real sextic curves on RP^2 with one real nondegenerate double point by I. Itenberg.

math.AG↗

arXiv2005

Real K3 surfaces with non-symplectic involution and applications. II

We consider real forms of relatively minimal rational surfaces F_m. Connected components of moduli of real non-singular curves in |-2K_{F_m}| had been classified recently for m=0, 1, 4 in math.AG/0312396. Applying similar methods, here we fill the gap for m=2 and m=3 to complete similar classification for any 0\le m\le 4 when |-2K_{F_m}| is reduced. The case of F_2 is especially remarkable and classical (quadratic cone in P^3). As an application, we finished classification of connected components of moduli of real hyper-elliptically polarized K3 surfaces and their deformations to real polarized K3 surfaces started in math.AG/0312396, math.AG/0507197. This could be important in some questions because real hyper-elliptically polarized K3 surfaces can be constructed explicitly.

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arXiv2003

Real K3 Surfaces with non-symplectic Involutions and Applications

Classification of real K3 surfaces X with a non-symplectic involution τis considered. For some exactly defined and one of the weakest possible type of degeneration (giving the very reach discriminant), we show that the connected component of their moduli is defined by the isomorphism class of the action of τand the anti-holomorphic involution ϕin the homology lattice. (There are very few similar cases known.) For their classification we apply invariants of integral lattice involutions with conditions which were developed by the first author in 1983. As a particular case, we describe connected components of moduli of real non-singular curves A\in |-2K_V| for the classical real surfaces: V=P^2, hyperboloid, ellipsoid, F_1, F_4. As an application, we describe all real polarized K3 surfaces which are deformations of general real K3 double rational scrolls (surfaces V above). There are very few exceptions. For example, any non-singular real quartic in P^3 can be constructed in this way.

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