Related papers: Canonical and log canonical thresholds of Fano com…

We show that the global log canonical threshold of generic Fano complete intersections of index 1 and codimension $k$ in ${\mathbb P}^{M+k}$ is equal to 1 if $M\geqslant 3k+4$ and the highest degree of defining equations is at least 8. This…

We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an application, we show that they have a Kaehler-Einstein metric if they are general.

We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain pro-jective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete…

We compute global log canonical thresholds of certain birationally bi-rigid Fano 3-folds embedded in weighted projective spaces as complete intersections of codimension 2 and prove that they admit an orbifold K\"{a}hler-Einstein metric and…

We prove that a Fano complete intersection of codimension $k$ and index 1 in the complex projective space ${\mathbb P}^{M+k}$ for $k\geqslant 20$ and $M\geqslant 8k\log k$ with at most multi-quadratic singularities is birationally…

We compute global log canonical thresholds of some smooth Fano threefolds.

We relate the global log canonical threshold of a variety with torus action to the global log canonical threshold of its quotient. We apply this to certain Fano varieties and use Tian's criterion to prove the existence of Kahler-Einstein…

In this paper we show that the global (log) canonical threshold of $d$-sheeted covers of the $M$-dimensional projective space of index 1, where $d\geqslant 4$, is equal to one for almost all families (except for a finite set). The varieties…

We investigate the variation of log canonical thresholds in (graded) linear systems. For toric log Fano varieties, we give a sharp lower bound for log canonical thresholds of the anticanonical members in terms of the global minimal log…

The purpose of this article is to develop techniques for estimating basis log canonical thresholds on logarithmic surfaces. To that end, we develop new local intersection estimates that imply log canonicity. Our main motivation and…

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…

We study Gorenstein almost Fano threefolds X with canonical singularities and pseudoindex > 1. We show that the maximal Picard number of X is 10 in general, 3 if X is Fano, and 8 if X is toric. Moreover, we characterize the boundary cases.…

We show that the optimal upper bound for the anticanonical degrees of non-Gorenstein $\mathbb{Q}$-factorial canonical Fano threefolds with Picard number one is 200/3.

Let $X$ be a smooth hypersurface of degree $n\geq 3$ in $\mathbb{P}^n$. We prove that the log canonical threshold of $H\in|-K_X|$ is at least $\frac{n-1}{n}$. Under the assumption of the Log minimal model program, we also prove that a…

Consider the Fano scheme $F_k(Y)$ parameterizing $k$-dimensional linear subspaces contained in a complete intersection $Y \subset \mathbb{P}^m$ of multi-degree $\underline{d} = (d_1, \ldots, d_s)$. It is known that, if $t := \sum_{i=1}^s…

We provide a complete classification of Fano threefolds X having canonical Gorenstein singularities and the anticanonical degree (-KX)^3 equal 64.

We prove that every smooth Fano complete intersection of index $1$ and codimension $r$ in $\mathbb{P}^{n+r}$ is birationally superrigid and K-stable if $n\ge 10r$. We also propose a generalization of Tian's criterion of K-stability and, as…

We consider the Fano scheme $F_k(X)$ of $k$--dimensional linear subspaces contained in a complete intersection $X \subset \mathbb{P}^n$ of multi--degree $\underline{d} = (d_1, \ldots, d_s)$. Our main result is an extension of a result of…

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 generalize Givental's Theorem for complete intersections in smooth toric varieties in the Fano case. In particular, we find Gromov--Witten invariants of Fano varieties of dimension $\geq 3$, which are complete intersections in weighted…