Related papers: Birationally rigid complete intersections of quadr…

We generalize the equivariant intermediate Jacobian torsor obstruction over $\mathbb{C}$ to algebraically closed fields of characteristic zero. It is an obstruction to the (projective) linearizability problem of finite group actions on…

We study three-dimensional Fano varieties with $\mathbb{C}^*$-action. Complementing recent results [13], we give classification results in the canonical case, where the maximal orbit quotient is $\mathbb{P}_2$ having a line arrangement of…

Given a hypersurface $i \colon X \hookrightarrow \widetilde{P}^n$ in a weighted projective space, we compute the intersection form on the second cohomology $H^2(X, \mathbb{Z})^{\otimes n-1} \to \mathbb{Z}$ for the purpose of identifying…

We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to…

It is known that a necessary condition for the existence of K\"ahler-Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian-Zhu. In a recent work of Berman-Nystr\"om, it was generalized for (singular) Fano…

We describe explicit birational maps from some rational complete intersections of three quadrics in $\mathbb{P}^7$ to some prime Fano manifolds together with their Sarkisov decomposition via a single Secant Flop, allowing us to recover the…

We explicitly describe the K-moduli compactifications and wall crossings of log pairs formed by a Fano complete intersection of two quadric threefolds and a hyperplane, by constructing an isomorphism with the VGIT quotient of such complete…

We classify three-dimensional nodal Fano varieties that are double covers of smooth quadrics branched over intersections with quartics acted on by finite simple non-abelian groups, and study their rationality.

We prove flexibility of two families of affine varieties: the complement in $\mathbb{P}^n$ of a projective quadric of rank at least three and affine cones over a smooth complete intersection of two quadrics in $\mathbb{P}^{n + 2}$, $n \ge…

We show a relation between the birational superrigidity of Fano manifold and its slope stability in the sense of Ross-Thomas.

We give a sufficient condition for birational superrigidity of del Pezzo fibrations of degree $1$ with only $\frac{1}{2} (1,1,1)$ singular points, generalizing the so called $K^2$-condition. As an application, we also prove that a del Pezzo…

We study a particular class of rationally connected manifolds, $X\subset \p^N$, such that two general points $x,x' \in X$ may be joined by a conic contained in $X$. We prove that these manifolds are Fano, with $b_2\leq 2$. Moreover, a…

We construct a $14$-dimensional affine variety $\Sigma^{14}_{\mathbb{A}}$ with a $\rm{GL}_3$- and a $(\mathbb{C}^*)^6$-actions. We denote by $\Sigma^{13}_{\mathbb{A}}$ the affine variety obtained from $\Sigma^{14}_{\mathbb{A}}$ by setting…

We give an explicit construction for the extension of a symmetric determinantal quartic K3 surface to a Fano 6-fold. Remarkably, the moduli of the 6-fold extension are in one-to-one correspondence with the moduli of the quartic surface. As…

We compute global log canonical thresholds, or equivalently alpha invariants, of birationally rigid orbifold Fano threefolds embedded in weighted projective spaces as codimension two or three. As an important application, we prove that most…

Let $V_1$ be the Fano threefold given as a hypersurface of degree 6 in $P(1,1,1,2,3)$ (over a number field $K$). Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points of $X$ is Zariski dense.

We study two rational Fano threefolds with an action of the icosahedral group $\mathfrak{A}_5$. The first one is the famous Burkhardt quartic threefold, and the second one is the double cover of the projective space branched in the Barth…

Let \phi: \mathbb{P}^{r}\dashrightarrow Z be a birational transformation with a smooth connected base locus scheme, where Z\subseteq\mathbb{P}^{r+c} is a nondegenerate prime Fano manifold. We call \phi a quadro-quadric special briational…

References to the works of Iliev-Ranestad and Kuznetsov added. ----- In a first part we detail the construction, on a general Fano 4-fold of genus 9, of a canonical set of four stable vector bundles of rank 2, and prove that they are rigid.…

Let $f\colon X\to Y$ be a surjective morphism of Fano manifolds of Picard number 1 whose VMRTs at a general point are not dual defective. Suppose that the tangent bundle $T_X$ is big. We show that $f$ is an isomorphism unless $Y$ is a…