New examples of cylindrical Fano fourfolds

International audienceWe produce new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form Z × A 1 , where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective G a-actions. Similar constructions of cylindrical Fano threefolds and fourfolds were done previously in [KPZ11, KPZ14, PZ15]

Genera of curves on a very general surface in P3P^3

In this paper we consider the question of determining the geometric genera of irreducible curves lying on a very general surface $S$ of degree $d$ at least 5 in $\mathbb{P}^3$ (the cases $d \leqslant 4$ are well known). We introduce the set $Gaps(d)$ of all non-negative integers which are not realized as geometric genera of irreducible curves on $S$. We prove that $Gaps(d)$ is finite and, in particular, that $Gaps(5)= \{0,1,2\}$. The set $Gaps(d)$ is the union of finitely many disjoint and separated integer intervals. The first of them, according to a theorem of Xu, is $Gaps_0(d):=[0, \frac{d(d-3)}{2} - 3]$. We show that the next one is $Gaps_1(d):= [\frac{d^2-3d+4}{2}, d^2-2d-9]$ for all $d \geqslant 6$.Comment: 16 page

Embeddings of ℂ*-surfaces into weighted projective spaces

Let V be a normal affine surface which admits a C*- and a C+-action. In this note we show that in many cases V can be embedded as a principal Zariski open subset into a hypersurface of a weighted projective space. In particular, we recover a result of D. Daigle and P. Russell

Smooth affine surfaces with non-unique C*-actions

In this paper we complete the classification of effective C*-actions on smooth affine surfaces up to conjugation in the full automorphism group and up to inversion of C*. If a smooth affine surface V admits more than one C*-action then it is known to be Gizatullin i.e., it can be completed by a linear chain of smooth rational curves. In our previous paper we gave a sufficient condition, in terms of the Dolgachev- Pinkham-Demazure (or DPD) presentation, for the uniqueness of a C*-action on a Gizatullin surface. In the present paper we show that this condition is also necessary, at least in the smooth case. In fact, if the uniqueness fails for a smooth Gizatullin surface V which is neither toric nor Danilov-Gizatullin, then V admits a continuous family of pairwise non-conjugated C*-actions depending on one or two parameters. We give an explicit description of all such surfaces and their C*-actions in terms of DPD presentations. We also show that for every k > 0 one can find a Danilov- Gizatullin surface V (n) of index n = n(k) with a family of pairwise non-conjugate C+-actions depending on k parameters

Hyperbolicity of generic deformations

10p. This text has intersection with another one (``Hyperbolicity of general deformations", arXiv:0709.2883v1 [math.AG]) of the same author. The difference is: the present one contains complete proofs, whereas the previous one is the content of the authors talk at the conference ``Effective Aspects of Complex Hyperbolic Varieties", Aber Wrac'h, France, September 10-14, 2007.International audienceWe modify the deformation method explored previously in a joint work of B. Shiffman and the author, in order to construct further examples of Kobayashi hyperbolic surfaces in the projective 3-space of any even degree starting with degree 8