Related papers: Some surfaces with non-polyhedral nef cones

Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the…

We investigate nef and movable cones of hypersurfaces in Mori dream spaces. The first result is: Let $Z$ be a smooth Mori dream space of dimension at least four whose extremal contractions are of fiber type of relative dimension at least…

We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in P^m x P^n, we describe their nef, movable and effective cones and determine when they are Mori dream spaces. Using these…

We prove that, for an Enriques surface in odd characteristic, the automorphism group is finitely generated and it acts on the effective nef cone with a rational polyhedral fundamental domain. We also construct a smooth projective surface in…

We show that if $X$ is a projective hyperk\"ahler fourfold and there exists a nonzero effective divisor $D$ which is not of bi-elliptic type and contained in the boundary of the nef cone of $X$, then $X$ contains a rational curve. This is a…

We study the different notions of semipositivity for (1,1) cohomology classes on K3 surfaces. We first show that every big and nef class (and every nef and rational class) is semiample, and in particular it contains a smooth semipositive…

The paper contains a general construction which produces new examples of non simply-connected smooth projective surfaces. We analyze the resulting surfaces and their fundamental groups. Many of these fundamental groups are expected to be…

We prove that a smooth surface, non of general type, in projective four-space, which lies on a quartic hypersurface with isolated singularities has degree at most 27 (in fact we prove a slightly more general result).

Let $X$ be a smooth complex projective rationally connected threefold with $-K_X$ nef and not semi-ample. In our previous work, we classified all such threefolds when $|{-}K_X|$ has no fixed divisor. In this paper, we continue our…

We show that the pseudo-effective cone of divisors of $\overline{M}_{0,n}$ is not polyhedral for $n\geq 8$ by constructing an extremal non-polyhedral ray of the dual cone of moving curves via maps on meromorphic strata of differentials…

Let $S$ be a non-uniruled (i.e., non-birationally ruled) smooth projective surface. We show that the tangent bundle $T_S$ is pseudo-effective if and only if the canonical divisor $K_S$ is nef and the second Chern class vanishes, i.e.,…

The nef cone of a projective variety Y is an important and often elusive invariant. In this paper we construct two polyhedral lower bounds and one polyhedral upper bound for the nef cone of Y using an embedding of Y into a toric variety.…

We describe the effective and the big cones of a projective symmetric variety. Moreover, we give a necessary and sufficient combinatorial criterion for the bigness of a nef divisor on a projective symmetric variety. When the variety is…

In continuation of our paper in Math. Ann. 333 we classify smooth complex projective threefolds X with -K_X big and nef but not ample and Picard number 2, whose anticanonical map is small. We assume also that the Mori contraction of X and…

In this paper we construct first examples of smooth projective surfaces of general type satisfying the following conditions: there are 1) an ample integral curve $C$ with $C^2=1$ and $h^0(X,O_X(C))=1$; \quad 2) a divisor $D$ with $(D,…

We propose a linear version of the weighted bounded negativity conjecture. It considers a smooth projective surface $X$ over an algebraically closed field of characteristic zero and predicts the existence of a common lower bound on…

Answering an old question, we find a domain X in the complex projective plane CP^2 which admits a strongly plurisubharmonic function, but such that every holomorphic function on X is constant. The domain X can be chosen diffeomorphic to an…

We recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the firdt nontrivial case of curves in RP^3. Namely, we show that i) the tangent developable surface of any convex…

We prove that a projective contact manifold X with second Betti number at least 2 whose canonical bundle K_X is not nef, is always the projectivised tangent bundle P(T_Y) of a projective manifold Y. It is expected that the canonical bundle…

We describe smooth rational projective algebraic surfaces X, over an algebraically closed field of characteristic different from 2, having an even set of four disjoint (-2)-curves N_1,...,N_4, i.e. such that N_1+...+N_4 is divisible by 2 in…