### $\mathbb{Q}$-Fano threefolds of large Fano index. I.

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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 a flat 3-dimensional...

In this paper we classify rank two Fano bundles $\mathcal{E}$ on Fano manifolds satisfying ${H}^{2}(X,\mathbb{Z})\cong {H}^{4}(X,\mathbb{Z})\cong \mathbb{Z}$. The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization $\mathbb{P}\left(\mathcal{E}\right)$, that allows us to obtain the cohomological invariants of $X$ and $\mathcal{E}$. As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.

I prove the algebraic stability and compute the dynamical degrees of C. Voisin’s rational self-map of the variety of lines on a cubic fourfold.

Let F=X-H:${k}^{n}$ → ${k}^{n}$ be a polynomial map with H homogeneous of degree 3 and nilpotent Jacobian matrix J(H). Let G=(G1,...,Gn) be the formal inverse of F. Bass, Connell and Wright proved in [1] that the homogeneous component of ${G}_{i}$ of degree 2d+1 can be expressed as ${G}_{i}^{\left(d\right)}={\sum}_{T}\alpha {\left(T\right)}^{-1}{\sigma}_{i}\left(T\right)$, where T varies over rooted trees with d vertices, α(T)=CardAut(T) and ${\sigma}_{i}\left(T\right)$ is a polynomial defined by (1) below. The Jacobian Conjecture states that, in our situation, $F$ is an automorphism or, equivalently, ${G}_{i}^{\left(d\right)}$ is zero for sufficiently large d....

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$\left(C\right){\left(\mathbb{R}\right)}^{0}$ 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$\left(C\right){\left(\mathbb{R}\right)}^{0}$. It generalizes the classical geometric description of the group law on the neutral component of the set of real points of...