Related papers: $\mathbb{Q}$-Fano threefolds with three birational…
We determine the rationality of very general quasismooth Fano 3-fold weighted hypersurfaces completely and determine the stable rationality of them except for cubic 3-folds. More precisely we prove that (i) very general Fano 3-fold weighted…
We prove that a weak $\mathbb{Q}$-Fano $3$-fold with terminal singularities has unobstructed deformations. By using this result and computing some invariants of a terminal singularity, we provide two results on global deformation of a weak…
In part I of this paper we constructed certain fibered Calabi-Yaus by a quotient construction in the context of weighted hypersurfaces. In this paper look at the case of K3 fibrations more closely and study the singular fibers which occur.…
We study Fano threefolds with~terminal singularities admitting a "minimal" action of a finite group. We prove that under certain additional assumptions such a variety does not contain planes. We also obtain an upper bounds of the number of…
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 paper explores the birational geometry of terminal quartic 3-folds. In doing this I develop a new approach to study maximal singularities with positive dimensional centers. This allows to determine the pliability of a Q-factorial…
We investigate birationality of the anti-pluricanonical map $\phi_{-m}$, the rational map defined by the anti-pluricanonical system $|-mK|$, on $\mathbb{Q}$-Fano threefolds.
Motivated by the general question of existence of open A1-cylinders in higher dimensional pro-jective varieties, we consider the case of Mori Fiber Spaces of relative dimension three, whose general closed fibers are isomorphic to the…
We prove that a nodal quartic threefold $X$ containing no planes is $Q$-factorial provided that it has not more than 12 singular points, with the exception of a quartic with exactly 12 singularities containing a quadric surface. We give…
Among smooth non-rational Fano 3-folds, the non-hyperelliptic Fano 3-fold X(10) of genus 6 has the unique property to admit a non-trivial orbit of birationally isomorphic 3-folds, inside its moduli space. Here we prove that these orbits…
We investigate birational properties of hypersurfaces of degree $6$ in the weighted projective space $\mathbf{P}(1,1,2,2,3)$. In particular, we prove that any such quasi-smooth hypersurface is not rational.
In this note we consider the problem of determining which Fano manifolds can be realised as fibres of a Mori fibre space. In particular, we study the case of toric varieties, Fano manifolds with high index and some Fano manifolds with high…
We classify birational maps into elliptic fibrations of a general quasismooth hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ that has terminal singularities.
In this study, we introduce a new class of rational elliptic 3-folds, which we refer to as "1/2 Calabi-Yau 3-folds". We construct elliptically fibered Calabi-Yau 3-folds by utilizing these rational elliptic 3-folds. The construction yields…
We classify Q-factorial Gorenstein Fano non-degenerate complete intersection threefolds in fake weighted projective spaces.
We consider the classification problem of prime $\mathbb{Q}$-Fano 3-folds with at most $1/2(1,1,1)$-singularities, which was initiated in [Taka2]. We construct two distinct classes of such 3-folds with genus one and six…
We study the birational geometry of hypersurfaces in products of weighted projective spaces, extending results previously established by J. C. Ottem. For most cases where these hypersurfaces are Mori dream spaces, we determine all relevant…
We prove that $\mathbb{Q}$-Fano threefolds of Fano index $\ge 8$ are rational.
We survey some results on the nonrationality and birational rigidity of certain hypersurfaces of Fano type. The focus is on hypersurfaces of Fano index one, but hypersurfaces of higher index are also discussed.
We classify Fano threefolds with only terminal singularities whose canonical class is Cartier and divisible by 2, and satisfying an additional assumption that the $G$-invariant part of the Weil divisor class group is of rank 1 with respect…