International Journal of Mathematics

In this paper, we give a definition of $\mathbb{Z}$-valued functions from the ambient isotopy classes of spherical/plane curves derived from chord diagrams, denoted by ${\sum }_i{\alpha }_i{x}_i$. Then, we introduce certain elements of the free $\mathbb{Z}$-module generated by the chord diagrams with at most $l$ chords, called relators of Type (I) ((SII), (WII), (SIII), or (WIII), respectively), and introduce another function ${\sum }_i{\alpha }_i{\tilde{x}}_i$ derived from ${\sum }_i{\alpha }_i{x}_i$. The main result (Theorem 1) shows that if ${\sum }_i{\alpha }_i{\tilde{x}}_i$ vanishes for the relators of Type (I) ((SII), (WII), (SIII), or (WIII), respectively), then ${\sum }_i{\alpha }_i{x}_i$ is invariant under the Reidemeister move of type RI (strong RII, weak RII, strong RIII, or weak RIII, respectively) that is defined in [N. Ito and Y. Takimura, $(1,2)$ and weak $(1,3)$ homotopies on knot projections, J. Knot Theory Ramifications 22 (2013) 1350085 14 pp].

Given any quadruple $(N,P,Q,M)$ of ${II}_1$-factors with finite index, the notions of interior and exterior angles between $P$ and $Q$ were introduced in [An angle between intermediate subfactors and its rigidity, Trans. Amer. Math. Soc. 371(8) (2019) 5973–5991]. We determine the possible values of these angles in terms of the cardinalities of the Weyl groups of the intermediate subfactors when $(N,P,Q,M)$ is an irreducible quadrilateral and the subfactors $N\subset P$ and $N\subset Q$ are both regular. For an arbitrary irreducible quadruple, an attempt is made to determine the values of angles by deriving expressions for the angles in terms of the common norm of two naturally arising auxiliary operators and the indices of the intermediate subfactors of the quadruple. Finally, certain bounds on angles between $P$ and $Q$ are obtained when $N\subset P$ is regular, which enforce some restrictions on the index of $N\subset Q$ in terms of that of $N\subset P$.

Let $X$ be a smooth complex projective curve of genus $g$ and let $L$ be a line bundle on $X$ with $degL>0$. Let $M$ be the moduli space of semistable rank 2 $L$-twisted Higgs bundles with trivial determinant on $X$. Let $M_X$ be the moduli space of stable rank 2 $L$-twisted Higgs bundles with determinant ${𝒪}(-x)$ for some $x\in X$ on $X$. We construct a cycle in the product of a stack of rational maps from nonsingular curves to $M_X$ and ${\mathrm{\text{Pic}}}^{2degL}(X)$ by using Hecke modifications of a stable $L$-twisted Higgs bundle in $M$.

We develop an algorithm detecting real representatives in equisingular strata of projective models of $K3$-surfaces. We apply this algorithm to spatial quartics and find two new examples of real strata without real representatives. As a by-product, we also give a new proof for the only previously known example of plane sextics.

The purpose of this work is to study a class of singular elliptic system involving the $p(x)$-Laplace operator of the form

Consider the moduli space of framed flat $U(2)$ connections with fixed odd determinant over a surface. Newstead combined some fundamental facts about this moduli space with the Mayer–Vietoris sequence to compute its betti numbers over any field not of characteristic 2. We adapt his method in characteristic 2 to produce conjectural recursive formulae for the mod two betti numbers of the framed moduli space which we partially verify. We also discuss the interplay with the mod two cohomology ring structure of the unframed moduli space.

The purpose of this paper is to explore a few properties of polynomial shift-like automorphisms of ${ℂ}^k.$ We first prove that a $\nu$-shift-like polynomial map (say $S_a$) degenerates essentially to a polynomial map in $\nu$-dimensions as $a\to 0.$ Second, we show that a shift-like map obtained by perturbing a hyperbolic polynomial (i.e. $S_a$, where $\left|a\right|$ is sufficiently small) has finitely many Fatou components, consisting of basins of attraction of periodic points and the component at infinity.

We define formal orbifolds over an algebraically closed field of arbitrary characteristic as curves together with some branch data. Their étale coverings and their fundamental groups are also defined. These fundamental groups approximate the fundamental group of an appropriate affine curve. We also define vector bundles on these objects and the category of orbifold bundles on any smooth projective curve. Analogues of various statements about vector bundles which are true in characteristic zero are also proved. Some of these are positive characteristic avatars of notions which appear in the second author’s work [A. J. Parmeswaran, Parabolic coverings I: Case of curves, J. Ramanujam Math. Soc. 25(3) (2010) 233–251.] in characteristic zero.

Let $𝕜$ be any field of characteristic zero, $X$ be a del Pezzo surface and $G$ be a finite subgroup in $Aut(X)$. In this paper, we study when the quotient surface $X/G$ can be non-rational over $𝕜$. Obviously, if there are no smooth $𝕜$-points on $X/G$ then it is not $𝕜$-rational. Therefore, under assumption that the set of smooth $𝕜$-points on $X/G$ is not empty we show that there are few possibilities for non-$𝕜$-rational quotients.

The quotients of del Pezzo surfaces of degree $2$ and greater are considered in the author’s previous papers. In this paper, we study the quotients of del Pezzo surfaces of degree $1$. We show that they can be non-$𝕜$-rational only for the trivial group or cyclic groups of order $2$, $3$ and $6$. For the trivial group and the group of order $2$, we show that both $X$ and $X/G$ are not $𝕜$-rational if the $G$-invariant Picard number of $X$ is $1$. For the groups of order $3$ and $6$, we construct examples of both $𝕜$-rational and non-$𝕜$-rational quotients of both $𝕜$-rational and non-$𝕜$-rational del Pezzo surfaces of degree $1$ such that the $G$-invariant Picard number of $X$ is $1$.

As a result of complete classification of non-$𝕜$-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of $𝕜$-rational surfaces, and obtain some corollaries concerning fields of invariants of $𝕜(x,y)$.