arXiv subjects
Publications and source records attributed to Konstantin Loginov.
We construct explicit unirational del Pezzo surfaces of degree $1$ with arithmetic Picard rank one over $\mathbb{Q}$, $\mathbb{F}_5$, and $\mathbb{C}(t)$. Moreover, we prove that every smooth real geometrically rational surface is unirational over $\mathbb{R}$ if and only if it has a real point.
Let $G=(\mathbb{Z}/4)^4$. We prove that if $X$ is a rationally connected threefold with a faithful action of $G$, then $X$ is $G$-birational to the Fermat quartic threefold. If $X$ is a terminal $G\mathbb{Q}$-Fano threefold, this birational equivalence is biregular. Consequently, the group $G$ acts faithfully on a rationally connected threefold but does not embed into $\operatorname{Cr}_3(\mathbb{C})$. Combined with earlier results, this yields a complete classification of the pairs $(m,r)$ for which $(\mathbb{Z}/m)^r$ embeds into $\operatorname{Cr}_3(\mathbb{C})$, and of those for which it embeds into $\operatorname{Bir}(X)$ for a rationally connected threefold $X$.
We compute the coregularity of del Pezzo surfaces with du Val singularities. To this aim, we study the relation between del Pezzo surfaces of degree $1$ and elliptic fibrations. It turns out that del Pezzo surfaces with positive coregularity correspond to isotrivial elliptic fibrations with some special properties. We also prove results about coregularity of del Pezzo surfaces over non-algebraically closed fields of characteristic $0$. Our results confirm the expectation that "most" del Pezzo surfaces have coregularity $0$, while del Pezzo surfaces with positive coregularity enjoy some special properties.
We introduce birational strong complete regularity and strong complete regularity, two numerical invariants for pairs of (relative) Fano type. They are defined using variants of qdlt Fano type models and the dimension of the dual complex of the reduced boundary, and can be viewed as Fano type refinements of Shokurov's complete regularity. We establish basic properties of these invariants and clarify its relation to models of qdlt Fano type appearing in K-stability. In particular, we prove that any pair with maximal birational strong complete regularity is $1$-complementary, and the thresholds where birational strong complete regularity or strong complete regularity jumps satisfy the ascending chain condition.
We study finite abelian groups acting on three-dimensional rationally connected varieties. We concentrate on the groups of K3 type, that is, abelian extensions by a cyclic group of groups that faithfully act on a K3 surface. In particular, if a finite abelian group faithfully acts on a threefold preserving a K3 surface (with at worst du Val singularities), then such a group is of K3 type. We prove a classification theorem for the groups of K3 type which can act on three-dimensional rationally connected varieties. We note the relation between certain groups of K3 type and K3 surfaces with higher Picard number.
We introduce and study the notion of $G$-coregularity of algebraic varieties endowed with an action of a finite group $G$. We compute $G$-coregularity of smooth del Pezzo surfaces of degree at least 6, and give a characterization of groups that can act on conic bundles with $G$-coregularity 0. We describe the relations between the notions of $G$-coregularity, $G$-log-canonical thresholds, $G$-rigidity, and exceptional quotient singularities.
We study the action of the group of bimeromorphic automorphisms $\mathrm{Bim}(X)$ of a compact complex manifold $X$ on the image of the pluricanonical map, which we call the projective pluricanonical representation of this group. If $X$ is a Moishezon variety, then the image of $\mathrm{Bim}(X)$ via such a representation is a finite group by a classical result due to Deligne and Ueno. We prove that this image is a finite group under the assumption that for the Kodaira dimension $κ(X)$ of $X$ we have $κ(X)=\dim X-1$. To this aim, we prove a version of the canonical bundle formula in relative dimension $1$ which works for a proper morphism from a complex variety to a projective variety. In particular, this establishes the analytic version of Prokhorov--Shokurov conjecture in relative dimension $1$. Also, we observe that the analytic version of this conjecture does not hold in relative dimension $2$.
We prove several boundedness results for log Fano pairs with certain K-stability. In particular, we prove that K-semistable log Fano pairs of Maeda type form a log bounded family. We also compute K-semistable domains for some examples.
We study the group of birational automorphisms of the $n$-dimensional projective space that preserve the standard torus invariant volume form with logarithmic poles. We prove that this group is not generated by pseudo-regularizable maps for $n\geq 4$ over $\mathbb{C}$, and for $n\geq 3$ over number fields. As a corollary, we show that this group is not simple in these cases.
We initiate the study of finite abelian groups that faithfully act on $3$-dimensional rationally connected varieties. We show that these groups can be naturally divided into three types: The groups of product type are finite abelian groups that are products of two groups that belong to the Cremona group of rank~$1$ and $2$, respectively; the groups of K3 type faithfully act on $G\mathbb{Q}$-Fano threefolds $X$ preserving a K3 surface $S\in|-K_X|$ with at worst du Val singularities; the third type consists of groups that act on $G\mathbb{Q}$-Fano threefolds with empty anti-canonical linear system. The classification of groups of product type follows from a result of J. Blanc. For the groups of K3 type, we establish a boundedness result. We also formulate a conjecture regarding the groups of the third type.
We prove that a general rational smooth Fano threefold admits a toric model. More precisely, for a general rational smooth Fano threefold $X$, we show the existence of a boundary divisor $D$ for which $(X,D)\simeq_{\rm cbir} (\mathbb{P}^3,H_0+H_1+H_2+H_3)$, where the $H_i$'s are the coordinate hyperplanes. In particular, a general rational smooth Fano threefold has birational complexity zero. We argue that the three conditions: rationality, generality, and smoothness are indeed necessary for the theorem.
We prove that all general smooth Fano threefolds of Picard rank $3$ and degree $14$ are K-stable, where the generality condition is stated explicitly.
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 family has coregularity 0; for the remaining 5 families, we obtain some partial results. In particular, we show that there exist families of smooth Fano threefolds whose general elements have positive coregularity.
Using the Abban-Zhuang theory and the classification of three-dimensional log smooth log Fano pairs due to Maeda, we prove that threefold log Fano pairs $(X, D)$ of Maeda type with reducible boundary $D$ are K-unstable, with four exceptions. We also correct several inaccuracies in Maeda's classification.
We prove that the image of the pluricanonical representation of a group of bimeromorphic automorphisms of a complex manifold has bounded finite subgroups. As a consequence, we show that the group of bimeromorphic automorphisms of an $n$-dimensional complex manifold whose Kodaira dimension is at least $n-2$, satisfies the Jordan property.
We prove that all smooth Fano threefolds of rank 4 and degree 24 are K-stable.
In this paper we investigate non-rationality of divisors on 3-fold log Fano fibrations $(X,B)\to Z$ under mild conditions. We show that if $D$ is a component of $B$ with coefficient $\ge t>0$ which is contracted to a point on $Z$, then $D$ is birational to $\mathbb{P}^1\times C$ where $C$ is a smooth projective curve with gonality bounded depending only on $t$. Moreover, if $t>\frac{1}{2}$, then genus of $C$ is bounded depending only on $t$.
We prove that a finite $3$-group in the Cremona group $\mathrm{Cr}_3(\mathbb{C})$ can be generated by at most $4$ elements. This provides the last missing piece in bounding the ranks of finite $p$-subgroups in the space Cremona group.