We investigate the following quasilinear and singular problem, $$\text{t}o2.7cm\left\{\begin{array}{cc}-{\Delta }_{p}u=\frac{\lambda }{u^{\delta }}+u^q\hfill & \text{in}\Omega ;\hfill \\ {u|}_{\partial \Omega }=0,u\>0\hfill & \text{in}\Omega ,\hfill \end{array}\right.\text{t}o2.7cm\text{(P)}$$ where $\Omega$ is an open bounded domain with smooth boundary, $1\<p\<\infty$, $p-1\<q\le p^{*}-1$, $\lambda \>0$, and $0\<\delta \<1$. As usual, $p^{*}=\frac{Np}{N-p}$ if $1\<p\<N$, $p^{*}\in (p,\infty )$ is arbitrarily large if $p=N$, and $p^{*}=\infty$ if $p\>N$. We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in $W_0^{1,p}\left(\Omega \right)$. While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle...

Let $C$ be a smooth curve of genus $g$. For each positive integer $r$ the birational $r$-gonality $s_r\left(C\right)$ of $C$ is the minimal integer $t$ such that there is $L\in {\text{Pic}}^t\left(C\right)$ with $h^0(C,L)=r+1$. Fix an integer $r\ge 3$. In this paper we prove the existence of an integer $g_r$ such that for every integer $g\ge g_r$ there is a smooth curve $C$ of genus $g$ with $s_{r+1}\left(C\right)/(r+1)>s_r\left(C\right)/r$, i.e. in the sequence of all birational gonalities of $C$ at least one of the slope inequalities fails.

Let $\mathrm{T}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta :=\{z\in ℂ:|z|<1\}$, normalized by $f\left(0\right)=f^{\text{'}}\left(0\right)-1=0$ and such that Im $z$ Im $f\left(z\right)$ $\ge 0$ for $z\in \Delta$. Moreover, let us denote: ${\mathrm{T}}^{\left(2\right)}:=\{f\in \mathrm{T}:f\left(z\right)=-f(-z)\text{for}z\in \Delta \}$ and ${\mathrm{T}}^{M,g}:=\{f\in \mathrm{T}:f\prec Mg\text{in}\Delta \}$, where $M>1$, $g\in \mathrm{T}\cap \mathrm{S}$ and $\mathrm{S}$ consists of all analytic functions, normalized and univalent in $\Delta$.We investigate  classes in which the subordination is replaced with the majorization and the function $g$ is typically real but does not necessarily univalent, i.e. classes $\{f\in \mathrm{T}:f\ll Mg\text{in}\Delta \}$, where $M>1$, $g\in \mathrm{T}$, which we denote...

Let $X$ be a proper, smooth, geometrically connected curve over a $p$-adic field $k$. Lichtenbaum proved that there exists a perfect duality: $$\text{Br}\left(X\right)\times \text{Pic}\left(X\right)\to \mathbb{Q}/\mathbb{Z}$$ between the Brauer and the Picard group of $X$, from which he deduced the existence of an injection of $\text{Br}\left(X\right)$ in ${\displaystyle \prod _{P\in X}\text{Br}\left(k_P\right)}$ where $P\in X$ and $k_P$ denotes the residual field of the point $P$. The aim of this paper is to prove that if $G=\tilde{G}$ is an $X_{et}$- scheme of semi-simple simply connected groups (s.s.s.c groups), then we can deduce from Lichtenbaum’s results...

We consider the problem $$\left\{\begin{array}{cc}-\Delta u=u^{\frac{N+2}{N-2}+\lambda }\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u\&gt;0\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u=0\hfill & \text{on}\partial \left(\Omega \setminus \epsilon \omega \right),\hfill \end{array}\right.$$ where $\Omega$ and $\omega$ are smooth bounded domains in ${ℝ}^N$, $N\ge 3$, $\epsilon \&gt;0$ and $\lambda \in ℝ.$ We prove that if the size of the hole $\epsilon$ goes to zero and if, simultaneously, the parameter $\lambda$ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}\left(V\right):=dim{H}^0(V,12K_V)\&gt;0$ and $P_{m_0}\left(V\right)\&gt;1$ for some positive integer $m_0\le 24$. A direct consequence is the birationality of the pluricanonical map ${\varphi }_m$ for all $m\ge 126$. Besides, the canonical volume $\text{Vol}\left(V\right)$ has a universal lower bound $\nu \left(3\right)\ge \frac{1}{63·{126}^2}$.

The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in ${\mathbb{P}}^3$. We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, $B$ and $E$, we give a necessary and sufficient condition for $B$ to be the branch curve of a surface $X$ in ${\mathbb{P}}^N$ and $E$ to be the image of the double curve of a ${\mathbb{P}}^3$-model of $X$. In the classical Segre theory, a...

Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1\cdots x_{n+1}\in I$ for $x_1,...,x_{n+1}\in R$, then there are $n$ of the $x_i$’s whose product is in $I$ and conjecture that ${\omega }_{R\left[X\right]}\left(I\left[X\right]\right)={\omega }_R\left(I\right)$ for any ideal $I$ of an arbitrary ring $R$, where ${\omega }_R\left(I\right)=min\{n:I\text{is}\text{an}n\text{-absorbing}\text{ideal}\text{of}R\}$. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following...

Let $E$ and $f$ be an Eisenstein series and a cusp form, respectively, of the same weight $k\ge 2$ and of the same level $N$, both eigenfunctions of the Hecke operators, and both normalized so that $a_1\left(f\right)=a_1\left(E\right)=1$. The main result we prove is that when $E$ and $f$ are congruent mod a prime $𝔭$ (which we take in this paper to be a prime of $\overline{\mathbb{Q}}$ lying over a rational prime $p\&gt;2$), the algebraic parts of the special values $L(E,\chi ,j)$ and $L(f,\chi ,j)$ satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions, ...

Let $C$ be a hyperelliptic curve of genus $g\ge 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\ge 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside...

Let $k$ be an algebraically closed field of characteristic $p\&gt;0$. We study obstructions to lifting to characteristic $0$ the faithful continuous action $\phi$ of a finite group $G$ on $k\left[\right[t\left]\right]$. To each such $\phi$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $\phi$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic $0$ in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H\subset G$. We determine for which $G$ the KGB...

We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k\ge 2$ and every set $A$ of words over $k$ satisfying $\lim {\sup }_{n\to \infty }|A\cap {\left[k\right]}^n|/k^n>0$ there exist a word $c$ over $k$ and a sequence $\left(w_n\right)$ of left variable words over $k$ such that the set $c\cup \{c^{}{w}_0{\left(a_0\right)}^{}..{.}^{}{w}_n\left(a_n\right):n\in \mathbb{N}\text{and}{a}_0,...,a_n\in \left[k\right]\}$ is contained in $A$. While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.

This paper classifies surfaces $S$ of general type with $p_g=q=1$ having an involution $i$ such that $S/i$ has non-negative Kodaira dimension and that the bicanonical map of $S$ factors through the double cover induced by $i.$ It is shown that $S/i$ is regular and either: a) the Albanese fibration of $S$ is of genus 2 or b) $S$ has no genus 2 fibration and $S/i$ is birational to a $K3$ surface. For case a) a list of possibilities and examples are given. An example for case b) with $K^2=6$ is also constructed.

We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring $R$. Our focus is on homological properties of contracting endomorphisms of $R$, e.g., the Frobenius endomorphism when $R$ contains a field of positive characteristic. For instance, in this case, when $R$ is $F$-finite and $C$ is a semidualizing $R$-complex, we prove that the following conditions are equivalent: (i) $C$ is a dualizing $R$-complex; (ii) $C\sim 𝐑{\mathrm{Hom}}_R{(}^{n}R,C)$ for some $n>0$; (iii) ${\mathrm{G}}_C{\text{-dim}}^{n}R<\infty$ and $C$ is derived...

Assume that $\mathrm{\Omega }$, ${\mathrm{\Omega }}^{\prime }$ are planar domains and $f:\mathrm{\Omega }\stackrel{\text{onto}}{\to }{\mathrm{\Omega }}^{\prime }$ is a homeomorphism belonging to Sobolev space $W_{\text{loc}}^{1,1}(\mathrm{\Omega };{ℝ}^2)$ with finite distortion. We prove that if the distortion function $K_f$ of $f$ satisfies the condition , then the distortion function $K_{f^{-1}}$ of $f^{-1}$ belongs to $L_{\text{loc}}^1({\mathrm{\Omega }}^{\prime })$. We show that this result is sharp in sense that the conclusion fails if ${\mathrm{dist}}_{\text{EXP}}(K_f,L^{\mathrm{\infty }})=1$. Moreover, we prove that if the distortion function $K_f$ satisfies the condition ${\mathrm{dist}}_{\text{EXP}}(K_f,L^{\mathrm{\infty }})=\lambda$ for some $\lambda >0$, then $K_{f^{-1}}$ belongs to $L_{\text{loc}}^p({\mathrm{\Omega }}^{\prime })$ for every $p\in (0,\frac{1}{2\lambda })$. As special case of this result we show that if...

We take up the study of the Brill-Noether loci $W^r(L,X):=\{\eta \in {\mathrm{Pic}}^0\left(X\right)|h^0(L\otimes \eta )\ge r+1\}$, where $X$ is a smooth projective variety of dimension $>1$, $L\in \mathrm{Pic}\left(X\right)$, and $r\ge 0$ is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for $h^0\left(K_D\right)$, where $D$ is a divisor that moves linearly on a smooth projective variety $X$ of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension $>2$. In the $2$-dimensional case...