Consider a projective variety X subset of 1n (over an algebraically closed field of characteristic zero), together with a (reduced) simple normal crossings divisor E subset of 1n, where the degrees of both X and E are at most d. We show there is a pair (n ', d ') which can be explicitly computed in terms of (n, d), such that (X, E) has a log resolution of singularities (X ', E '), where (X ', E ') can be embedded in 1n ' and both X ' and E ' have degrees at most d ' in 1n '.
We bound the indices of normal abelian subgroups in finite groups contained in the Cremona group of rank 2 over a field of odd characteristic.
We prove the base point free theorem for log canonical foliated pairs of rank one on a Q-factorial projective klt threefold. Moreover, we show abundance in the case of numerically trivial log canonical foliated pairs of rank one in any dimension.
In this short note we will explore some recent connections between positivity, singularities, and boundedness in various contexts focusing on birational geometry.
We give an effective upper bound for the index of klt complements on toric Fano varieties.
We construct examples of families of pairs over a DVR of positive characteristic, with very mild singularities, such that the MMP on the closed fiber does not extend to a relative MMP.
We revisit the recent theory of Sun-Zhang on general Fano fibrations which emerged from the study of non-compact K & auml;hler-Ricci soliton metrics [SZ24], primarily from an algebrogeometric perspective. In addition to reviewing the existing framework, we present new results, conjectures, and remarks. These include methods for computing weighted volumes via (restricted) volumes, Laplace transforms, and incomplete Gamma-functions, and a conjectural algebro-geometric construction ("bubbling") of Fano fibration with asymptotically conical base from degenerating Fano fibrations.
We prove compatibility relations between mixed Hodge numbers of k-Du Bois fibers in flat projective families and versal deformations of isolated k-Du Bois singularities. These extend the notion of polarized relations in asymptotic Hodge theory beyond the normal-crossing boundary case, and we study combinatorial properties of the resulting weak polarized relations graphs.
We study linearizability of actions of finite groups on cubic threefolds with non-isolated singularities.
We prove that the total Cartier index of a bounded family of projective varieties of klt type is bounded.
In characteristic zero, we construct logarithmic resolution of singularities, with simple normal crossings exceptional divisor, using weighted blow-ups.
We show that the anti-canonical bundle of any Qfactorial surface is numerically effective if and only if it is pseudo-effective. To prove this, we establish a numerical non-vanishing theorem for surfaces polarized with pseudo-effective divisors. The latter answers a question of C. Fontanari.
An extremal curve germ is a germ of a threefold X with terminal singularities along a connected reduced complete curve C such that there exists a KX-negative contraction f : X -> Z with C being a fiber. We give a rough classification of extremal curve germs with reducible central curve C.
Rationality is not a constructible property in families. In this article, we consider stronger notions of rationality and study their behavior in families of Fano varieties. We first show that being toric is a constructible property in families of Fano varieties. The second main result of this article concerns an intermediate notion that lies between toric and rational varieties, namely cluster type varieties. A cluster type & Qopf;-factorial Fano variety contains an open dense algebraic torus, but the variety does not need to be endowed with a torus action. We prove that, in families of & Qopf;-factorial terminal Fano varieties, being of cluster type is a constructible condition. As a consequence, we show that there are finitely many smooth families parametrizing n-dimensional smooth cluster type Fano varieties.
We study integral plane curves meeting at a single unibranch point and show that such curves must satisfy two equivalent conditions. A numeric condition: the local invariants of the curves at the contact point must be arithmetically related. A geometric condition: the tropical curves that we associate to the contact point must be isomorphic. Moreover, we prove closed formulas for the delta-invariant of a unibranch singularity, and for the dimension of the loci of curves with an assigned unibranch point. Our work is motivated by interest in the Lang exceptional set.
We prove several results relating the nonvanishing and the existence of good minimal models of different pairs that have the same underlying variety.
Let X be a compact Kahler manifold. We study subgroups G < Aut(X) of biholomorphic automorphisms of zero entropy when Aut(0)(X) is compact (e.g. when Aut(0)(X) is trivial). We show that the virtual derived length l(vir)(G) of G satisfies l(vir)(G) <= dim X - kappa(X), where kappa(X) is the Kodaira dimension of X. Modulo the main conjecture of our previous work concerning the essential nilpotency class, we obtain the same upper bound c(vir)(G) <= dim X - kappa(X) for the virtual nilpotency class c(vir)(G), together with a geometric description of G (sic) X when the equality holds.
In this paper, we classify simple smooth modules over the superconformal current algebra $\frak g$. More precisely, we first classify simple smooth modules over the Heisenberg-Clifford algebra, and then prove that any simple smooth $\frak g$-module is a tensor product of such modules for the super Virasoro algebra and the Heisenberg-Clifford algebra, or an induced module from a simple module over some finite-dimensional solvable Lie superalgebras. As a byproduct, we provide characterizations for both simple highest weight $\frak g$-modules and simple Whittaker $\frak g$-modules. Additionally, we present several examples of simple smooth $\frak g$-modules that are not tensor product of modules over the super Virasoro algebra and the Heisenberg-Clifford algebra.
