The main result of this paper is as follows: let $X,Y$ be smooth projective threefolds (over a field of characteristic zero) such that $b_2\left(X\right)=b_2\left(Y\right)=1$. If $Y$ is not a projective space, then the degree of a morphism $f:X\to Y$ is bounded in terms of discrete invariants of $X$ and $Y$. Moreover, suppose that $X$ and $Y$ are smooth projective $n$-dimensional with cyclic Néron-Severi groups. If $c_1\left(Y\right)=0$, then the degree of $f$ is bounded iff $Y$ is not a flat variety. In particular, to prove our main theorem we show the non-existence of...

First we find effective bounds for the number of dominant rational maps $f:X\to Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type $\{A·K_X^n{\}}^{\{B·K_X^n{\}}^2}$, where $n=\dim X$, $K_X$ is the canonical bundle of $X$ and $A,B$ are some constants, depending only on $n$. Then we show that for any variety $X$ there exist numbers $c\left(X\right)$ and $C\left(X\right)$ with the following properties: For any threefold $Y$ of general type the number of dominant rational maps $f:X\to Y$ is bounded above by $c\left(X\right)$. ...

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.

Let $C$ be a smooth real quartic curve in ${\mathbb{P}}^2$. Suppose that $C$ has at least $3$ real branches $B_1,B_2,B_3$. Let $B=B_1\times B_2\times B_3$ and let $O\in B$. Let ${\tau }_O$ be the map from $B$ into the neutral component Jac$\left(C\right){\left(ℝ\right)}^0$ of the set of real points of the jacobian of $C$, defined by letting ${\tau }_O\left(P\right)$ be the divisor class of the divisor $\sum P_i-O_i$. Then, ${\tau }_O$ is a bijection. We show that this allows an explicit geometric description of the group law on Jac$\left(C\right){\left(ℝ\right)}^0$. It generalizes the classical geometric description of the group law on the neutral component of the set of real...

Sia $X$ una varietà abeliana e $L$ un fibrato in rette ampio di tipo $\left(2,2d_2,\mathrm{\dots },2d_g\right)$ su $X$; sia ${\phi }_L$ l'applicazione associata a $L$. In questo lavoro si dimostra il seguente fatto: se $d_i\le 2$ per qualsiasi $i$, $L$ non è mai normalmente generato (quindi, se ${\phi }_L$ è un embedding, ${\phi }_L\left(X\right)$ non è proiettivamente normale); negli altri casi invece $L$ è normalmente generato per $\left(X,c_1\left(L\right)\right)$ generico nello spazio dei moduli delle varietà abeliane polarizzate di tipo $\left(2,2d_2,\mathrm{\dots },2d_g\right)$.

Let $Y$ be an integral projective curve with $g:=p_a\left(Y\right)\ge 2$. For all positive integers $d$, $r$ let $W_d^r\left(Y\right)\left({}^{*A*}\right)$ be the set of all $L\in {\text{Pic}}^d\left(Y\right)$ with $h^0\left(Y,L\right)\ge r+1$ and $L$ spanned. Here we prove that if $d\le g-2$, then $dim\left(W_d^r\left(Y\right)\left({}^{*A*}\right)\right)\le d-3r$ except in a few cases (essentially if $Y$ is a double covering).