Related papers: Quarto-quartic birational maps of $\mathbb{P}_3(\m…

For $q\geq 3$, we let $\mathcal{S}_q$ denote the projectivization of the set of symmetric $q\times q$ matrices with coefficients in $\mathbb{C}$. We let $I(x)=(x_{i,j})^{-1}$ denote the matrix inversion, and we let $J(x)=(x_{i,j}^{-1})$ be…

We prove the existence of various families of irreducible homaloidal hypersurfaces in projective space $\mathbb P^ r$, for all $r\geq 3$. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared…

Given a smooth genus three curve $C$, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in $\mathbb{P}^8$ as a hypersurface whose singular locus is the Kummer threefold of $C$; this hypersurface is the…

We study a coarse moduli space of irreducible representations of the group of unipotent matrices of order $\mathbb{4}$ over the ring of integers which have finite weight. All such representations are known to be monomial. To describe a…

In this letter we give fourth-order autonomous recurrence relations with two invariants, whose degree growth is cubic or exponential. These examples contradict the common belief that maps with sufficiently many invariants can have at most…

We study the notion of a birational Chow-K\"unneth decomposition, which is essentially a decomposition of the integral birational motive of a variety. The existence of a birational Chow-K\"unneth decomposition is stably birationally…

We study and classify up to a linear conjugation germs of flows in the space of quadratic birational transformations of the complex projective space of dimension 3. As a consequence we show that every quadratic flow preserves a pencil of…

This is a continuation of "Rational families of vector bundles on curves, I". Let C be a smooth projective curve of genus at least 2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1.…

We consider the space $P$ of generic complex 5-degree polynomials. Critical values of such polynomial, i.e. four points in the complex plane, either are vertices of a convex quadrangle $Q$, or vertices of a triangle $T$ with one point…

We derive a formula for the generating function of d-irreducible bipartite planar maps with several boundaries, i.e. having several marked faces of controlled degrees. It extends a formula due to Collet and Fusy for the case of arbitrary…

Special birational transformations $\Phi:\p^r\da Z$ defined by quadric hypersurfaces are studied by means of the variety of lines $\mathcal L_z\subset\p^{r-1}$ passing through a general point $z\in Z$. Classification results are obtained…

We study transformations as in the title with emphasis on those having smooth connected base locus, called "special". In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by…

We give an explicit description of the irreducible components of the moduli spaces of polarized Enriques surfaces in terms of decompositions of the polarization as an effective sum of isotropic classes. We prove that infinitely many of…

We consider a new class of quaternionic mappings, associated with the spatial partial differential equations. We describe all mappings from this class using four analytic functions of the complex variable.

This paper is devoted to the complete algebraic and geometric classification of complex 4 and 5-dimensional antiassociative algebras. In particular, we proved that the variety of complex 4-dimensional antiassociative algebras has dimension…

Let V^{r}_{d,g, \delta} be the Hilbert scheme of nodal curves in P^r of degree d and arithmetic genus g with \delta nodes. Under suitable numerical assumptions on d and g, for every 0 \le \delta \le g we construct an irreducible component…

We prove the existence of three irreducible curves $C_{12,m}$ of degree 12 with the same number of cusps and different Alexander polynomials. This exhibits a Zariski triple. Moreover we provide a set of generators for the elliptic threefold…

We develop a theory of $p$-adic continued fractions for a quaternion algebra $B$ over $\mathbb Q$ ramified at a rational prime $p$. Many properties holding in the commutative case can be proven also in this setting. In particular, we focus…

We give explicit equations that describe the character variety of the figure eight knot for the groups SL(3,C), GL(3,C) and PGL(3,C). This has five components of dimension 2, one consisting of totally reducible representations, another one…

We classify the irreducible components of the Hilbert scheme of $n$ points on non-reduced algebraic plane curves, and give a formula for the multiplicities of the irreducible components. The irreducible components are indexed by partitions…