Let $f \colon S \rightarrow B$ be a fibred surface over an algebraically closed field $k$ of characteristic $p \geq 5$, where $S$ is a minimal smooth projective surface of general type and $B$ is a smooth projective curve of genus $b\geq 2$. We prove that the group of fibration-preserving automorphisms of $f$ has order at most $15658(K_S^2)^{4}$. Furthermore, we provide an example to show that the exponent $4$ of the polynomial bound is sharp.
Let $k$ be an algebraically closed field of characteristic $p>0$, and let $S$ be a minimal smooth projective surface of general type over $k$. If the second Chern class satisfies $c_2(S)<0$, then the order of the automorphism group satisfies $|\mathrm{Aut}_k(S)|<2598(K_S^2)^4$, and the order of every abelian subgroup $...
Let $X$ be an algebraic variety of dimension $d$ over an algebraically closed field $k$ of zero characteristic. Suppose that $G$ is a virtually polycyclic subgroup in the group $\mathrm{Bir}(X)$ of birational automorphisms of $X$. We show that the virtual derived length of $G$ does not exceed $2d+1$. Moreover, if $X$ i...
Let $L$ be a line bundle of negative degree on an elliptic curve $E$. While the ruled surface $M := \mathbb P(\mathcal O_E \oplus L)$ is one of the simplest varieties in algebraic geometry, its tangent bundle $T_M$ is surprisingly complex: we first show that the ring of symmetric tensors is not finitely generated. We t...
Let $f\colon X\to X$ be a polarised endomorphism of a complex projective manifold, and let $D\subset X$ be a reduced totally invariant hypersurface. We prove that every irreducible component of the singular locus of $D$ has codimension one in $D$. In particular, if $D$ is normal, then it is smooth. As an application we...
For each $g\ge 2$, we show that there are finitely many cyclic commensurability classes of closed, fibered arithmetic hyperbolic $3$-manifolds with genus $g$ fiber. More generally, we show that there are only finitely many conjugacy classes of admissible surface subgroups of PSL$_2(\mathbb{C})$ whose image is contained...
Tamunonye Cheetham-West, Ho-Min Lee, N. Miller· 1 citation· ⚡1
We study hyperelliptic curves on $K3$ surfaces. We show that any number of general points on a $K3$ surface of Picard rank one can be connected by a hyperelliptic curve. As a by-product we recover the Beauville--Voisin decomposition of the small diagonal, together with some new bounds on its torsion order in the integr...
Federico Moretti· 0 citations
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