Let k be an algebraically closed field of characteristic 0. We give a brief survey on multiplicity-2 structures on varieties. Let $Z$ be a reduced irreducible nonsingular $(n-1)$-dimensional variety such that $2Z = X \cap F$, where $X$ is a normal $n$-fold with canonical singularities, $F$ is an $(N-1)$-fold in $\mathbb{P}^N$, such that $Z \cap \mathrm{Sing}(X) \neq \emptyset$. Assume that $\mathrm{Sing}(X)$ is equidimensional and $\mathrm{codim}_X(\mathrm{Sing}(X)) = 3$. We study the singularities of $X$ through which $Z$ passes. We also consider Fano cones. We discuss the construction of some vector bundles and the resolution property of a variety.

This is a survey paper about studies of the simplest shape of the apparent contour for stable maps between surfaces. Such studies first appeared in [10] then in [1], [3], [6], [20], [22]. Let $M$ be a connected and closed surface, $N$ a connected surface. For a stable map $\varphi: M\to N$, denote by $c(\varphi)$, $n(\varphi)$ and $i(\varphi)$ the numbers of cusps, nodes and singular set components of $\varphi$, respectively. For a $C^\infty$ map $\varphi_0 : M\to S^2$ into the sphere, we study the minimal pair $(i, c+n)$ and triples $(i,c,n)$, $(c,i,n)$, $(n,c,i)$ and $(i,n,c)$ among stable maps $M\to S^2$ homotopic to $\varphi_0$ with respect to the lexicographic order.

L'objectif de cet article est de présenter quelques conjectures liées à l'étude des valuations divisorielles et la connexité en codimension 1, et de donner le lien entre ces conjectures et l'existence de certaines log-résolutions des singularités.

We complete the enumeration of the possible roots of the Alexander polynomial (both conventional and over finite fields) of a trigonal curve. The curves are not assumed proper or irreducible.

We show that the monodromy operator action on the first cohomology group of the Milnor fiber is combinatorially determined for line arrangements with at most triple points and containing at most 18 lines, with one possible exception. This last case depends on the existence of a line arrangement involving 18 lines and 48 triple points with a special distribution with respect to smooth conics.

Let $f(\mathbf{z},\bar{\mathbf{z}})$ be a polar weighted homogeneous mixed polynomial. If $f(\mathbf{z}, \bar{\mathbf{z}})$ has an isolated singularity at the origin $\mathbf{o}$, then $f(\mathbf{z}, \bar{\mathbf{z}})$ gives a fibered link in a sphere centered at $\mathbf{o}$. In this paper, we study fibered links which are determined by polar weighted homogeneous mixed polynomials and show the existence of mixed polynomials whose Milnor fibers cannot be obtained from a disk by plumbings of Hopf bands.

Let $C$ be a nonsingular curve on a rational surface $S$. In the case when the logarithmic 2 genus of $C$ is equal to two, Iitaka proved that the geometric genus of $C$ is either zero or one and classified such pairs $(S, C)$. In this article, we prove the existence of these classes with geometric genus one in Iitaka's classification. The curve in the class is a singular curve on $\mathbb{P}^2$ or the Hirzebruch surface $\Sigma_d$ and its singularities are not in general position. For this purpose, we provide the arrangement of singular points by considering invariant curves under a certain automorphism of $\Sigma_d$.

We show the existence of infinitely many links of non-degenerate simple $K3$ singularities defined by non-quasi-homogeneous polynomials such that the second betti numbers of the links are 17, which do not appear in the case of the singularities defined by quasi-homogeneous polynomials.

Let $C = \{f = 0\}$ be an affine plane curve. In this paper, we study $(4,3)$ line degenerations of torus curves. Line degenerations of torus curves are divided into two types which are called visible or invisible degenerations. We will show that there does not exist a $(4,3)$ line degenerated torus curve which has two types decompositions.

In this paper, a new effective algorithm for computing algebraic local cohomology classes associated with semi-quasihomogeneous singularities, is presented. The key ingredients of the proposed algorithm are weighted-degrees and Poincaré polynomials for algebraic local cohomology. An extension of the algorithm to parametric cases is also discussed.

The paper shows certain geometric-arithmetic aspects with some new observations in consideration of variables separated curves, e.g. relationships with generalized Chebyshev polynomials, Chebyshev pencils, local variant at infinity of Stothers–Mason $abc$-theorem, Stothers–Langevin pairs, Pell–Abel conics (or polynomial Pell equations), Belyi maps, etc. Several case studies and open problems are discussed.

Let $f(\mathbf{z}, \bar{\mathbf{z}})$ be a mixed polynomial with strongly non-degenerate face functions. We consider a canonical toric modification $\pi: X\to\mathbb{C}^n$ and a polar modification $\pi_{\mathbb{R}}: Y\to X$. We will show that the toric modification resolves topologically the singularity of $V$ and the zeta function of the Milnor fibration of $f$ is described by a formula of a Varchenko type.

Takamura [Ta3] established a theory of splitting families of degenerations of complex curves of genus $g \ge 1$. He introduced a powerful method for constructing a splitting family, called a barking family, in which the resulting family of complex curves has a singular fiber over the origin (the main fiber) together with other singular fibers (subordinate fibers). He made a list of barking families for genera up to 5 and determined the main fibers appearing in them. This paper determines most of the subordinate fibers of the barking families in Takamura's list for the case $g = 1$. (There remain four undetermined cases.) Also, we show that some splittings never occur in a splitting family.

We report on our recent study on the mirror symmetry between orbifold projective lines and cusp singularities. Both of their homological mirror symmetry, an equivalence of triangulated categories, and their classical mirror symmetry, an isomorphism of Frobenius manifolds, are discussed.