Let $X$ be a uniruled projective manifold and let $x$ be a general point. The main result of [2] says that if the $(-K_X)$-degrees (i.e., the degrees with respect to the anti-canonical bundle of $X$) of all rational curves through $x$ are at least $dimX+1$, then $X$ is a projective space. In this paper, we study the structure of $X$ when the $(-K_X)$-degrees of all rational curves through $x$ are at least $dimX$. Our study uses the projective variety ${𝒞}_x\subset \mathbb{P}{T}_x\left(X\right)$, called the VMRT at $x$, defined as the union of tangent directions to the...

Let $k$ be an algebraically closed field of characteristic $p\>0$. 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...

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.

Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of $X_0^{+}\left(p^r\right)\left(\mathbb{Q}\right)$, for $r\>1$ and $p$ a prime number exceeding $2·{10}^{11}$. This includes the case of the curves $X_{\mathrm{split}}\left(p\right)$. We then prove, with the help of computer calculations, that the same holds true for $p$ in the range $11\le p\le {10}^{14}$, $p\ne 13$. The combination of those results completes the qualitative study of rational points on $X_0^{+}\left(p^r\right)$ undertook in our previous work, with the only exception of $p^r={13}^2$.

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 $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}\left(V\right):=dim{H}^0(V,12K_V)\>0$ and $P_{m_0}\left(V\right)\>1$ 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 $\nu \left(3\right)\ge \frac{1}{63·{126}^2}$.

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...

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...

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO ${\Sigma }_1$, ${\Sigma }_2$, ${\Sigma }_3$, ${\Sigma }_4$ INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................

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 ${\rho }_X-{\rho }_T=4$.