Let $Y$ be an $(m+1)$-dimensional irreducible smooth complex projective variety embedded in a projective space. Let $Z$ be a closed subscheme of $Y$, and $\delta$ be a positive integer such that ${ℐ}_{Z,Y}\left(\delta \right)$ is generated by global sections. Fix an integer $d\ge \delta +1$, and assume the general divisor $X\in |H^0(Y,{ℐ}_{Z,Y}\left(d\right))|$ is smooth. Denote by $H^m{(X;\mathbb{Q})}_{\perp Z}^{\mathrm{van}}$ the quotient of $H^m(X;\mathbb{Q})$ by the cohomology of $Y$ and also by the cycle classes of the irreducible components of dimension $m$ of $Z$. In the present paper we prove that the monodromy representation on $H^m{(X;\mathbb{Q})}_{\perp Z}^{\mathrm{van}}$ for the family...

The relative cohomology ${\mathrm{H}}_{\mathrm{diff}}^1(𝕂\left(1\right|3),\mathrm{𝔬𝔰𝔭}(2,3);{𝒟}_{\lambda ,\mu }\left(S^{1|3}\right))$ of the contact Lie superalgebra $𝕂\left(1\right|3)$ with coefficients in the space of differential operators ${𝒟}_{\lambda ,\mu }\left(S^{1|3}\right)$ acting on tensor densities on $S^{1|3}$, is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating $1$-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$-cocycle $s\left(X_f\right)=D_1{D}_2{D}_3\left(f\right){\alpha }_3^{1/2}$, $X_f\in 𝕂\left(1\right|3)$ which is invariant with respect to the conformal subsuperalgebra $\mathrm{𝔬𝔰𝔭}(2,3)$ of $𝕂\left(1\right|3)$. In this work we study the supergroup case. We give an explicit construction of $1$-cocycles...

Let $G$ be a split semisimple linear algebraic group over a field and let $T$ be a split maximal torus of $G$. Let $𝗁$ be an oriented cohomology (algebraic cobordism, connective $K$-theory, Chow groups, Grothendieck’s $K_0$, etc.) with formal group law $F$. We construct a ring from $F$ and the characters of $T$, that we call a formal group ring, and we define a characteristic ring morphism $c$ from this formal group ring to $𝗁(G/B)$ where $G/B$ is the variety of Borel subgroups of $G$. Our main result says that when the...

Let $F=(f_1,...,f_q)$ be a polynomial dominating map from ${ℂ}^n$ to ${ℂ}^q$. We study the quotient ${𝒯}^1\left(F\right)$ of polynomial 1-forms that are exact along the generic fibres of $F$, by 1-forms of type $\mathrm{d}R+\sum a_i\mathrm{d}{f}_i$, where $R,a_1,...,a_q$ are polynomials. We prove that ${𝒯}^1\left(F\right)$ is always a torsion $ℂ[t_1,...,t_q]$-module. Then we determine under which conditions on $F$ we have ${𝒯}^1\left(F\right)=0$. As an application, we study the behaviour of a class of algebraic $({ℂ}^p,+)$-actions on ${ℂ}^n$, and determine in particular when these actions are trivial.

Let $R$ be a Noetherian ring, and $I$ and $J$ be two ideals of $R$. Let $S$ be a Serre subcategory of the category of $R$-modules satisfying the condition $C_I$ and $M$ be a $ZD$-module. As a generalization of the $S$-$\mathrm{depth}(I,M)$ and $\mathrm{depth}(I,J,M)$, the $S$-$\mathrm{depth}$ of $(I,J)$ on $M$ is defined as $S$-$\mathrm{depth}(I,J,M)=inf\{S$-$\mathrm{depth}(𝔞,M):𝔞\in \tilde{\mathrm{W}}(I,J)\}$, and some properties of this concept are investigated. The relations between $S$-$\mathrm{depth}(I,J,M)$ and $H_{I,J}^i\left(M\right)$ are studied, and it is proved that $S$-$\mathrm{depth}(I,J,M)=inf\{i:H_{I,J}^i\left(M\right)\notin S\}$, where $S$ is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology...

In this paper, we examine a natural question concerning the divisors of the polynomial $x^n-1$: “How often does $x^n-1$ have a divisor of every degree between $1$ and $n$?” In a previous paper, we considered the situation when $x^n-1$ is factored in $\mathbb{Z}\left[x\right]$. In this paper, we replace $\mathbb{Z}\left[x\right]$ with ${𝔽}_p\left[x\right]$, where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$.

Let $(R,𝔪)$ be a commutative Noetherian local ring, $𝔞$ be an ideal of $R$ and $M$ a finitely generated $R$-module such that $𝔞M\ne M$ and $\mathrm{cd}(𝔞,M)-\mathrm{grade}(𝔞,M)\le 1$, where $\mathrm{cd}(𝔞,M)$ is the cohomological dimension of $M$ with respect to $𝔞$ and $\mathrm{grade}(𝔞,M)$ is the $M$-grade of $𝔞$. Let $D(-):={\mathrm{Hom}}_R(-,E)$ be the Matlis dual functor, where $E:=E(R/𝔪)$ is the injective hull of the residue field $R/𝔪$. We show that there exists the following long exact sequence $$\begin{array}{cc}\hfill 0⟶& H_{𝔞}^{n-2}\left(D\left(H_{𝔞}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^n\left(D\left(H_{𝔞}^n\left(M\right)\right)\right)⟶D\left(M\right)\\ \hfill ⟶& H_{𝔞}^{n-1}\left(D\left(H_{𝔞}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^{n+1}\left(D\left(H_{𝔞}^n\left(M\right)\right)\right)\\ \hfill ⟶& H_{𝔞}^n\left(D\left(H_{(x_1,...,x_{n-1})}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^n\left(D\left(H_{(}^{n-1}M\right)\right))⟶...,\end{array}$$ where $n:=\mathrm{cd}(𝔞,M)$ is a non-negative integer, $x_1,...,x_{n-1}$ is a regular sequence in $𝔞$ on $M$ and, for an $R$-module $L$, $H_{𝔞}^i\left(L\right)$ is the $i$th local cohomology module...

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...

We consider the linear system $|2{\Theta }_0|$ of second order theta functions over the Jacobian $JC$ of a non-hyperelliptic curve $C$. A result by J.Fay says that a divisor $D\in |2{\Theta }_0|$ contains the origin $𝒪\in JC$ with multiplicity $4$ if and only if $D$ contains the surface $C-C=\left\{𝒪\right(p-q)\mid p,q\in C\}\subset JC$. In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing $𝒪$ with multiplicity $6$, divisors containing the fourfold $C_2-C_2=\{𝒪(p+q-r-s)\mid p,q,r,s\in C\}$, and divisors singular along $C-C$, using...

Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$. We consider the image ${𝒩}_1(D,X)$ of ${𝒩}_1\left(D\right)$ in ${𝒩}_1\left(X\right)$ under the natural push-forward of $1$-cycles. We show that ${\rho }_X-{\rho }_D\le codim{𝒩}_1(D,X)\le 8$. Moreover if $codim{𝒩}_1(D,X)\ge 3$, then either $X\cong S\times T$ where $S$ is a Del Pezzo surface, or $codim{𝒩}_1(D,X)=3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that ${\rho }_X-{\rho }_T=4$.

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\>2$ and equal to the braid group of the sphere $ℂ$ $P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

Let $ℐ$ be a coherent subsheaf of a locally free sheaf $𝒪\left(E_0\right)$ and suppose that $ℱ=𝒪\left(E_0\right)/ℐ$ has pure codimension. Starting with a residue current $R$ obtained from a locally free resolution of $ℱ$ we construct a vector-valued Coleff-Herrera current $\mu$ with support on the variety associated to $ℱ$ such that $\phi$ is in $ℐ$ if and only if $\mu \phi =0$. Such a current $\mu$ can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed....

Let $𝔞$, $I$, $J$ be ideals of a Noetherian local ring $(R,𝔪,k)$. Let $M$ and $N$ be finitely generated $R$-modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $H_{I,J}^t\left(M\right)$ and $D\left(H_{I,J}^t\left(M\right)\right)$, where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and $D(-):={\mathrm{Hom}}_R(-,E_R\left(k\right))$ is the Matlis dual functor. We show that if $R$ is a $d$-dimensional complete Cohen-Macaulay...

CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...

Let $\Delta$ be a pure simplicial complex on the vertex set $\left[n\right]=\{1,...,n\}$ and $I_{\Delta }$ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,...,x_n]$. We show that $\Delta$ is a matroid (complete intersection) if and only if $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$) is clean for all $m\in \mathbb{N}$ and this is equivalent to saying that $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$, respectively) is Cohen-Macaulay for all $m\in \mathbb{N}$. By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S/I^m$ or $S/I^{\left(m\right)}$ is not (pretty) clean for all integer $m\ge 3$. If $dim\left(\Delta \right)=1$, we also prove that $S/I_{\Delta }^{\left(2\right)}$ ($S/I_{\Delta }^2$) is clean if and only...

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.