Related papers: Coregularity of Fano varieties

The modular properties of some higher dimensional varieties of special Fano type are analyzed by computing the L-function of their $\Omega-$motives. It is shown that the emerging modular forms are string theoretic in origin, derived from…

Iterating the procedure of making a double cover over a given variety, we construct large families of smooth higher-dimensional Fano varieties of index 1. These varieties can be realized as complete intersections in various weighted…

In this short note we show the unboundedness of the dimension of the K-moduli space of $n$-dimensional Fano varieties, and that the dimension of the stack can also be unbounded while, simultaneously, the dimension of the corresponding…

We study the coregularity of smooth Fano threefolds. We prove that for 100 out of 105 families of smooth Fano threefolds, a general member in the family has coregularity 0; moreover, for 92 families out of these 100, any member in the…

We study the birational boundedness of special fibers of log Calabi-Yau fibrations and Fano fibrations. We show that for a locally stable family of Fano varieties or polarised log Calabi-Yau pairs over a curve, if the general fiber…

The Eisenbud--Goto conjecture states that $\operatorname{reg} X\le\operatorname{deg} X -\operatorname{codim} X+1$ for a nondegenerate irreducible projective variety $X$ over an algebraically closed field. While this conjecture is known to…

Complete intersections inside rational homogeneous varieties provide interesting examples of Fano manifolds. For example, if $X = \cap_{i=1}^r D_i \subset G/P$ is a general complete intersection of $r$ ample divisors such that $K_{G/P}^*…

We construct exceptional Fano varieties with the smallest known minimal log discrepancies in all dimensions. These varieties are well-formed hypersurfaces in weighted projective space. Their minimal log discrepancies decay doubly…

For Fano varieties of various singularities such as canonical and terminal, we construct examples with large Fano index. By low-dimensional evidence, we conjecture that our examples have the largest Fano index for all dimensions.

Let $X$ be a Fano type variety and $(X,\Delta)$ be a log Calabi-Yau pair with $\Delta$ a Weil divisor. If $(X,\Delta)$ admits a polarized endomorphism, then we show that $(X,\Delta)$ is a finite quotient of a toric pair. Along the way, we…

In this paper we investigate the degrees of irrationality of degenerations of $\epsilon$-lc Fano varieties of arbitrary dimensions. We show that given a generically $\epsilon$-lc klt Fano fibration $X\to Z$ of dimension $d$ over a smooth…

Given a Schubert variety X_w, we exhibit a divisor \Delta, defined over the integers, such that the pair (X_w,\Delta) is log Fano in all characteristics.

For a birational log Fano contraction, it is conjectured an inequality between the dimension of its exceptional locus and the minimal log discrepancy over the locus. The conjecture follows from the existence of the flip for the contraction…

We give conditions for a uniruled variety of dimension at least 2 to be non-solid. This study provides further evidence to a conjecture by Abban and Okada on the solidity of Fano 3-folds. To complement our results we write explicit…

A projective log variety (X, D) is called "a log Fano manifold" if X is smooth and if D is a reduced simple normal crossing divisor on X with -(K_X+D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this…

The complexity of a pair $(X,B)$ is an invariant that relates the dimension of $X$, the rank of the group of divisors, and the coefficients of $B$. If the complexity is less than one, then $X$ is a toric variety. We prove that if the…

In this paper, we develop an algebraic K-stability theory (e.g. special test configuration theory and optimal destabilization theory) for log Fano $\mathbb R$-pairs, and construct a proper K-moduli space to parametrize K-polystable log Fano…

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 give a brief survey of the concept of birational rigidity, from its origins in the two-dimensional birational geometry, to its current state. The main ingredients of the method of maximal singularities are discussed. The principal…

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index e, then the degree of irrationality of a very general complex Fano hypersurface of index e and dimension n…