相关论文: Fano's inequality is also false for three-dimensio…
An example of a Cremona transformation of the tree-dimensional projective space is presented. For the transformation, the inequalities of Fano are not valid.
A Cremona transformation of five-dimensional projective space is constructed. The degree of the transformation is 7. The inequalities of Fano are not fulfilled for this transformation.
This is the unabridged web version of the paper that will be published on the American Journal of Mathematics. In this paper, we study the birational geometry of certain examples of mildly singular quartic 3-folds. A quartic 3-fold is an…
We announce a factorization result for equivariant birational morphisms between toric 4-folds whose source is Fano: such a morphism is always a composite of blow-ups along smooth invariant centers. Moreover, we show with a counterexample…
Within an axiomatic framework for three-dimensional projective space based on lines alone, we explore the Fano axiom of harmonicity according to which the diagonal lines of a complete quadrilateral are not concurrent.
We characterize the birational geometry of some hyperk\"ahler fourfolds of Picard rank $3$ obtained as the Fano varieties of lines on cubic fourfolds containing pairs of cubic scrolls. In each of the two cases considered, we identify all of…
We give several examples of pairs of non-isomorphic cubic fourfolds whose Fano varieties of lines are birationally equivalent (and in one example isomorphic). Two of our examples, which are special families of conjecturally irrational…
In this note we study Fano threefolds with noncyclic torsion in the divisor class group. Since they can all be obtained as quotients of Fano threefolds, we get also all examples that can be obtained as quotients of low codimension Fanos in…
An analytical derivation of Fano line-shape asymmetry ratio has been presented here for a general case. It is shown that Fano line-shape becomes less asymmetric as \q is increased and finally becomes completely symmetric in the limiting…
We develop an equivariant version of the Pfaffian-Grassmannian correspondence and apply it to produce examples of nontrivial twisted equivariant stable birationalities between cubic threefolds and degree 14 Fano threefolds.
We survey what is known about Fano threefold weighted complete intersections from the point of view of birational rigidity.
It is demonstrated that, unless the meaning of conformal transformations for the underlying geometrical structure is discussed on a same footing as it is done for the equations of the given gravity theory, the notion of "conformal…
We show that, unlike del Pezzo surfaces, higher dimensional Fano manifolds do not satisfy in general boundedness properties for their ${\rm CH}_0$ group of $0$-cycles. For example, for quartic threefolds having a point of odd degree, there…
We construct an example of the birationally rigid complete intersection of a quadric and a cubic in $\PA^5$ with an ordinary double point, which under a small deformation gives a non-rigid Fano variety. Thus we show that birational rigidity…
We give a simple proof of the non-rationality of the Fano threefold defined by the equations \Sigma x_i = \Sigma x_i^2 = \Sigma x_i^3 = 0 in P^6 .
We prove birational superrigidity of generic Fano complete intersections $V$ of type $2^{k_1}\cdot 3^{k_2}$ in the projective space ${\mathbb P}^{2k_1+3k_2}$, under the condition that $k_2\geq 2$ and $k_1+2k_2=\mathop{\rm dim} V\geq 12$,…
We describe the Fano scheme of lines on a general cubic threefold containing a plane over a field $k$ of characteristic different from 2. Then, we use the Fano scheme to characterize rationality for such cubic threefolds over nonclosed…
We construct an example of a birational transformation of a rational threefold for which the first and second dynamical degrees coincide and are $>1$, but which does not preserve any holomorphic (singular) foliation. In particular, this…
We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.
Canonical transformation in a three-dimensional phase space endowed with Nambu bracket is discussed in a general framework. Definition of the canonical transformations is constructed as based on canonoid transformations. It is shown that…