Sia $X\subset {\mathbf{P}}^N$ una varietà irriducibile $n$-dimensionale localmente Cohen-Macaulay, $\mathbf{Q}$-Gorenstein e non di tipo generale; assumiamo $N=6$, $2n=N+2$ e $\text{dim}\left(\text{Sing}\left(X\right)\right)=2n-N$. In questo lavoro dimostriamo che $\text{deg}\left(X\right)\le {\left(N+1\right)}^{N-n}$ e quindi che l'insieme di tutte queste varietà è parametrizzato da un insieme finito di varietà algebriche.

A category of Brauer diagrams, analogous to Turaev’s tangle category, is introduced, a presentation of the category is given, and full tensor functors are constructed from this category to the category of tensor representations of the orthogonal group O$\left(V\right)$ or the symplectic group Sp$\left(V\right)$ over any field of characteristic zero. The first and second fundamental theorems of invariant theory for these classical groups are generalised to the category theoretic setting. The major outcome is that we obtain presentations...

Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $Q$, and let $\overline{X}$ and $\overline{Y}$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$, respectively, by extension of the ground field. We show that the Galois invariant subgroup of Br $\left(\overline{X}\right)\oplus$ Br($\overline{Y})$ has finite index in the Galois invariant subgroup of Br$(\overline{X}\times \overline{Y})$. This implies that the cokernel of the natural map Br$\left(X\right)\oplus$ Br$\left(Y\right)\to$ Br$(X\times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer–Manin set of the product of...

Let C be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic zero. For a fixed line bundle L on C, let M C (r; L) be the coarse moduli space of semistable vector bundles E over C of rank r with ∧r E = L. We show that the Brauer group of any desingularization of M C(r; L) is trivial.

Let $X$ be an algebraic variety defined over a field $k$ of characteristic $0$, and let $Y$ be an $X$-torsor under a torus. We compute the Brauer group of $Y$. In the case of a number field $k$ we deduce results concerning the arithmetic of $X$.

Let $X\to 𝒜$ be a non-constant morphism from a curve $X$ to an abelian variety $𝒜$, all defined over a number field $k$. Suppose that $X$ is a counterexample to the Hasse principle. We give sufficient conditions for the failure of the Hasse principle on $X$ to be accounted for by the Brauer–Manin obstruction. These sufficiency conditions are slightly stronger than assuming that $𝒜\left(k\right)$ and $Ш(𝒜/k)$ are finite.

The Briançon-Skoda number of a ring $R$ is defined as the smallest integer k, such that for any ideal $I\subset R$ and $l\ge 1$, the integral closure of $I^{k+l-1}$ is contained in $I^l$. We compute the Briançon-Skoda number of the local ring of any analytic irreducible planar curve in terms of its Puiseux characteristics. It turns out that this number is closely related to the Milnor number.