Related papers: Birational geometry of rational quartic surfaces

Two birational projective varieties in $P^n$ are Cremona Equivalent if there is a birational modification of $P^n$ mapping one onto the other. The minimal Cremona degree of $X\subset P^n$ is the minimal integer among all degrees of…

Two divisors in $\P^n$ are said to be Cremona equivalent if there is a Cremona modification sending one to the other. We produce infinitely many non equivalent divisorial embeddings of any variety of dimension at most 14. Then we study the…

Two divisors in $\mathbb P^n$ are said to be Cremona equivalent if there is a Cremona modification sending one to the other. In this paper I study irreducible cones in $\mathbb P^n$ and prove that two cones are Cremona equivalent if their…

Two reduced projective schemes are said to be Cremona equivalent if there is a Cremona map that maps one in the other. In this paper I revise some of the known results about Cremona equivalence and extend the main result of [MP09] to…

Are Fourier-Mukai equivalent cubic fourfolds birationally equivalent? We obtain an affirmative answer to this question for very general cubic fourfolds of discriminant 20, where we produce birational maps via the Cremona transformation…

Two projective varieties are said to be Cremona equivalent if there is a Cremona modification sending one onto the other. In the last decade, Cremona equivalence has been investigated widely, and we now have a complete theory for…

In this paper we consider the birational classification of pairs (S,L), with S a rational surfaces and L a linear system on S. We give a classification theorem for such pairs and we determine, for each irreducible plane curve B, its…

Let $X$ be a projective variety of dimension $r$ over an algebraically closed field. It is proven that two birational embeddings of $X$ in $\P^n$, with $n\geq r+2$ are equivalent up to Cremona transformations of $\P^n$.

A Cremona transformation is a birational self-map of the projective space $ \mathbb{P}^{n} $. Cremona transformations of $ \mathbb{P}^{n} $ form a group and this group has a rational action on subvarieties of $ \mathbb{P}^{n} $ and hence on…

We exhibit a Cremona transformation of ${\bf P}^4$ such that the base loci of the map and its inverse are birational to K3 surfaces. The two K3 surfaces are derived equivalent but not isomorphic to each other. As an application, we show…

We show that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in ${\mathbf P}^3$ such that $S_1$ and $S_2$ are isomorphic as abstract varieties but not Cremona isomorphic. We also show, in a geometrically explicit way,…

We show that even dimensional Fermat cubic hypersurfaces are rational over any field of characteristic different from three by producing explicit rational parametrizations given by polynomials of low degree. As a byproduct of our…

For any $n\geq 3$, we prove that there exist equivalences between these apparently unrelated objects: irreducible $n$-dimensional non degenerate projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal scrolls and…

One studies Cremona monomial maps by combinatorial means. Among the results is a simple integer matrix theoretic proof that the inverse of a Cremona monomial map is also defined by monomials of fixed degree, and moreover, the set of…

We study the quasi-projective variety Bir_d of plane Cremona transformations defined by three polynomials of fixed degree d and its subvariety Bir_d^o where the three polynomials have no common factor. We compute their dimension and the…

We show that the hessian map of quartic plane curves is a birational morphism onto its image, thus bringing new evidence for a very interesting conjecture of Ciro Ciliberto and Giorgio Ottaviani. Our new approach also yields a simpler proof…

We prove analogues of several well-known results concerning rational morphisms between quadrics for the class of so-called quasilinear $p$-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric…

This paper is concerned with suitable generalizations of a plane de Jonqui\`eres map to higher dimensional space $\mathbb{P}^n$ with $n\geq 3$. For each given point of $\mathbb{P}^n$ there is a subgroup of the entire Cremona group of…

We show by finding an explicit parametrization that a 4th degree surface which arises as a necessary condition for the existence of a perfect cuboid is a rational surface, i.e. birationally equivalent over $\mathbb Q$ to a plane.

A finite, normal cover $f: X\longrightarrow \bbP^2$ of degree $m\geq 3$ (the case $m=2$ is well known and we do not consider it in this paper) is called \emph{simple}, if there is a pencil $\mathcal P$ of rational curves of $\bbP^2$ such…