Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Real logarithmic models for real analytic foliations in the plane

Abstract: Let S be a germ of a holomorphic curve at (ℂ2,0) with finitely many branches S 1,…,S r and let ${\mathcal{I}}=(I_{1},\ldots,I_{r}) \in {\mathbb{C}}^{r}$ . We show that there exists a non-dicritical holomorphic foliation of logarithmic type at 0∈ℂ2 whose set of separatrices is S and having index I i along S i in the sense of Lins Neto (Lecture Notes in Math. 1345, 192–232, 1988) if the following (necessary) condition holds: after a reduction of singularities π:M→(ℂ2,0) of S, the vector ${\mathcal{I}}$ gives rise, by the usual rules of transformation of indices by blowing-ups, to systems of indices along components of the total transform $\bar{S}$ of S at points of the divisor E=π −1(0) satisfying: (a) at any singular point of $\bar{S}$ the two indices along the branches of $\bar{S}$ do not belong to ℚ≥0 and they are mutually inverse; (b) the sum of the indices along a component D of E for all points in D is equal to the self-intersection of D in M. This construction is used to show the existence of logarithmic models of real analytic foliations which are real generalized curves. Applications to real center-focus foliations are considered.

A Note on a Broken Dirichlet Convolution

Abstract: The paper deals with a broken Dirichlet convolution ⊗ which is based on using the odd divisors of integers. In addition to presenting characterizations of ⊗-multiplicative functions we also show an analogue of Menon’s identity: ∑ a (mod n) (a,n)⊗=1 (a− 1, n) = φ⊗(n)[τ(n)− 1 2 τ2(n)], where (a, n)⊗ denotes the greatest common odd divisor of a and n, φ⊗(n) is the number of integers a (mod n) such that (a, n)⊗ = 1, τ(n) is the number of divisors of n, and τ2(n) is the number of even divisors of n.

Singular rationally connected threefolds with non-zero pluri-forms

Abstract: This paper is concerned with singular projective rationally connected threefolds $X$ which carry non-zero pluri-forms, \textit{i.e.} $H^0(X,(\Omega_X^1)^{[\otimes m]}) eq \{0\}$ for some $m > 0$, where $(\Omega_X^1)^{[\otimes m]}$ is the reflexive hull of $(\Omega_X^1)^{\otimes m}$. If $X$ has $\mathbb{Q}$-factorial terminal singularities, then we show that there is a fibration $p$ from $X$ to $\mathbb{P}^1$. Moreover, there is a natural isomorphism from $H^0(X, (\Omega_X^1)^{[\otimes m]})$ to $H^0(\mathbb{P}^1, \mathscr{O}_{\mathbb{P}^1}(-2m+\sum_{z\in \mathbb{P}^1} [\frac{(m(p,z)-1)m}{m(p,z)}]))$ for all $m>0$, where $m(p,z)$ is the smallest positive coefficient in the divisor $p^*z$.

On the class of caustic on the moduli space of odd spin curves

Abstract: Let $C$ be a smooth projective curve of genus $g\geq 3$ and let $\eta$ be an odd theta characteristic on it such that $h^0(C,\eta) = 1$. Pick a point $p$ from the support of $\eta$ and consider the one-dimensional linear system $|\eta + p|$. In general this linear system is base-point free and all its ramification points (i.e. ramification points of the corresponding branched cover $C\to\mathbb P^1\simeq \mathbb PH^0(C,\eta+p)$) are simple. We study the locus in the moduli space of odd spin curves where the linear system $|\eta + p|$ fails to have this general behavior. This locus splits into a union of three divisors: the first divisor corresponds to the case when $|\eta+p|$ has a base point, the second one corresponds to theta characteristics which are not reduced at $p$ (and therefore $|\eta + p|$ must have a triple point at $p$) and the third one corresponds to the case when $|\eta + p|$ has a triple point different from $p$. The second divisor was studied by G. Farkas and A. Verra in\cite{FARo} where its expansion in the rational Picard group was used to prove that the moduli space of odd spin curves is of general type for genus at least $12$. We call the first divisor a Base Point divisor and the third one a Caustic divisor (following Arnold terminology for Hurwitz spases). The objective of this paper is to expand these two divisors via the set of standard generators in the rational Picard group of the moduli space of odd spin curves.