Related papers: Explicit Birational Geometry of Fano threefold com…
It is well known that there are totally 130 deformation families of quasi-smooth terminal weighted hypersurface Fano threefolds and all members belonging to 95 families of Fano indices one are birationally rigid. Among remaining $35$…
We construct Q-factorial terminal Fano varieties, starting in dimension 4, whose nef cone jumps when the variety is deformed. It follows that de Fernex and Hacon's results on deformations of 3-dimensional Fanos are optimal. The examples are…
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.
A known conjecture of Grinenko in birational geometry asserts that a Mori fibre space with the structure of del Pezzo fibration of low degree is birationally rigid if and only if its anticanonical class is an interior point in the cone of…
In our previous research, we constructed the affine varieties $\Sigma_{\mathbb{A}}^{13}$ and $\Pi_{\mathbb{A}}^{14}$ whose partial projectivizations admit $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibrations with relative Picard number one. In…
We consider the moduli space of rigidified perfect complexes with support on a general complete intersection Calabi-Yau threefold $X$ and its Tyurin degeneration $X\rightsquigarrow X_1\cup_SX_2$ to a complete intersection of Fano threefolds…
We study GIT stability of divisors in products of projective spaces. We first construct a finite set of one-parameter subgroups sufficient to determine the stability of the GIT quotient. In addition, we characterise all maximal orbits of…
We construct families of birational involutions on $\mathbb{P}^3$ or a smooth cubic threefold which do not fit into a non-trivial elementary relation of Sarkisov links. As a consequence, we construct new homomorphisms from their group of…
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…
It is known that all degenerations of the complex projective plane into a surface with only quotient singularities are controlled by the positive integer solutions $(a,b,c)$ of the Markov equation $$x^2+y^2+z^2=3xyz.$$ It turns out that…
It has been shown by Hacking and Prokhorov that if the projective surface X with quotient singularities and self-intersection number 9 has a smoothing to the projective plane, then X is the general fiber of a Q-Gorenstein deformation of the…
We study genus 1 Gromov-Witten invariants of Fano complete intersections in the projective spaces. Among other things, we show a reconstruction theorem for genus 1 invariants with only ambient insertions, and compute the genus 1 invariants…
We study the deformation theory of a Fano variety X with normal crossing singularities of dimension at most three. We obtain a formula for the sheaf T^1(X) of first order deformations of X in a suitable log resolution of X and its singular…
We shall study the structure of hyperelliptic fibrations of genus 3, from the view point given by Catanese and Pignatelli in arXiv:math/0503294. In this part I, we shall give a structure theorem for such fibrations for the case of f : S \to…
We classify Q-factorial Gorenstein Fano non-degenerate complete intersection threefolds in fake weighted projective spaces.
We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We…
We study congruences of lines $X_\omega$ defined by a sufficiently general choice of an alternating 3-form $\omega$ in $n+1$ dimensions, as Fano manifolds of index $3$ and dimension $n-1$. These congruences include the…
Let $X$ be a complex smooth Fano variety of dimension $n$. Assume that $X$ admits a birational contraction of an extremal ray. In this paper, we give a classification of such $X$ when the pseudoindex is equal to $\frac{\dim X}{2}$.
We propose a new method to study birational maps between Fano varieties based on multiplier ideal sheaves. Using this method, we prove equivariant birational rigidity of four Fano threefolds acted on by the group A6. As an application, we…
We classify Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3-folds with canonical Gorenstein…