Related papers: Rationality of Mukai varieties over non-closed fie…

Over a field $k$, we study rational UFDs of finite transcendence degree $n$ over $k$. We classify such UFDs $B$ when $n=2$, $k$ is algebraically closed, and $B$ admits a positive $\mathbb{Z}$-grading, showing in particular that $B$ is…

We use birational geometry to show that the existence of rational points on proper rationally connected varieties over fields of characteristic $0$ is a consequence of the existence of rational points on terminal Fano varieties. We discuss…

According to Mukai, any prime Fano threefold X of genus 7 is a linear section of the spinor tenfold in the projectivized half-spinor space of Spin(10). It is proven that the moduli space of stable rank-2 vector bundles with Chern classes…

In this note we describe a unique linear embedding of a prime Fano 4-fold F of genus 10 into the Grassmannian G(3,6). We use this to construct some moduli spaces of bundles on linear sections of F. In particular the moduli space of bundles…

We prove that $\mathbb{Q}$-Fano threefolds of Fano index $\ge 8$ are rational.

Let X be a complex Fano manifold of dimension n. Let s(X) be the sum of l(R)-1 for all the extremal rays of X, the edges of the cone NE(X) of curves of X, where l(R) denotes the minimum of (-K_X \cdot C) for all rational curves C whose…

Let X be a Fano variety of dimension n, pseudoindex i_X and Picard number \rho_X. A generalization of a conjecture of Mukai says that \rho_X(i_X-1)\le n. We prove that the conjecture holds if: a) X has pseudoindex i_X \ge \frac{n+3}{3} and…

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…

A complex smooth prime Fano threefold $X$ of genus $9$ is related via projective duality to a quartic plane curve $\Gamma$. We use this setup to study the restriction of rank $2$ stable sheaves with prescribed Chern classes on $X$ to an…

The aim of the present paper is to prove the rationality of the universal family of polarized $ K3 $ surfaces of degree 14. This is achieved by identifying it with the moduli space of cubic fourfolds plus the data of a quartic scroll. The…

We construct an explicit complex smooth Fano threefold with Picard number 1, index 1, and degree 10 (also known as a Gushel-Mukai threefold) and prove that it is not rational by showing that its intermediate Jacobian has a faithful…

Let $\mathcal{F}_{8,2\times \frac{1}{2}(1,1,1)}$ be the moduli space of genus $8$ Fano $3$-folds with two singular points of type $\frac{1}{2}(1,1,1)$. We show that $\mathcal{F}_{8,2\times \frac{1}{2}(1,1,1)}$ is a rational variety.

Let $X\subset P^n$ be a complex projective manifold of degree $d$ and arbitrary dimension. The main result of this paper gives a classification of such manifolds (assumed moreover to be connected, non-degenerate and linearly normal) in case…

We study the varieties of reductions associated to the four Severi varieties, the first example of which is the Fano threefold of index 2 and degree 5 studied by Mukai and others. We prove that they are smooth but very special linear…

Let $X$ be the Fano threefold of index one, degree $22$, and $\mathrm{Pic}(X)\cong\mathbb{Z}$. Such a threefold $X$ can be realized by a regular zero section $\mathbf{s}$ of $(\bigwedge^2\mathcal{F}^{*})^{\oplus 3}$ over Grassmannian…

We study the connectedness of the real locus of smooth geometrically rational Fano threefolds and prove a sufficient criterion of $\mathbb{R}$-rationality.

Mukai proved that most prime Fano fourfolds of degree 10 and index 2 are contained in a Grassmannian G(2,5). They are all unirational and some are rational, as remarked by Roth in 1949. We show that their middle cohomology is of K3 type and…

We prove the Mukai conjecture on the characterisation of products of projective spaces among Fano varieties for a class of locally factorial Fano varieties defined in terms of their Cox rings. The Fano varieties of this class are…

As a generalization of the Mukai conjecture, we conjecture that the Fano manifolds $X$ which satisfy the property $\rho_X(r_X-1)\geq\dim X-1$ have very special structure, where $\rho_X$ is the Picard number of $X$ and $r_X$ is the index of…

Let $M_{g, n}$ (respectively, $\overline{M_{g, n}}$) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to…