Related papers: On the Kodaira vanishing theorem for log del Pezzo…

The existence of a vector field on a compact Kaehler manifold with nonempty zero locus and the properties of this zero locus strongly influence the geometry of the manifold. For example, J. Wahl proved that the existence of a vector field…

The main result is that a quasi-projective surface has negative log Kodaira dimension (i.e. no log pluricanonical sections) iff it is dominated by images of the affine line. This follows from our main intermediate result, that the smooth…

We prove that there exists a nonzero holderian real-to-real function vanishing together with its M. Rietz potential in all points of some set of positive length. This result improves the one of D. Beliaev and V. Havin. We also extend the…

We construct a klt del Pezzo surface which is not globally F-split, over any algebraically closed field of positive characteristic.

We investigate the emptiness of adjoint linear systems associated to successive multiples of a given positive divisor with real coefficients

Let $Z \to X$ be a finite branched Galois cover of normal projective geometrically integral varieties of dimension $d \geq 2$ over a perfect field $k$. For such a cover, we prove a Chebotarev-type density result describing the decomposition…

In this paper we study the problem of existence of orbifold Kaehler-Einstein metrics on del Pezzo surfaces of degree 1 with Du Val singular points. Moreover we compute global log canonical thresholds of del Pezzo surfaces of degree 1 with…

Consider a log canonical pair $(X,B)$ such that there is a Cartier divisor $D$ for which $T_X(-\log B) \otimes \mathcal O(D)$ is locally free and globally generated. Let $\mathcal F$ be a log canonical foliation of rank 1 on $X$. We prove…

Making use of Gruson-Raynaud's technique of "platification par eclatement", Kerz and Strunk proved that the negative homotopy K-theory groups of a Noetherian scheme X of Krull dimension d vanish below -d. In this note, making use of…

Let Z be a general surface in P^3 of degree at least 5. Using a Lefschetz pencil argument, we give an elementary new proof of the vanishing of a regulator on K_1(Z).

We prove a non-vanishing theorem of the cohomology H^0 of the adjoint divisor K_X + L' where L' is the round up of a nef and big Q-divisor L.

Using certain Thom spectra appearing in the study of cobordism categories, we show that the odd half of the Miller-Morita-Mumford classes on the mappping class group of a surface with negative Euler characteristic vanish in integral…

Log del Pezzo surfaces play the role of the opposite of surfaces of general type. We will completely classify all the log del Pezzo surfaces of rank 2 and Cartier index 3 with a unique singularity.

We prove new local inequality for divisors on surfaces and utilize it to compute $\alpha$-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$,…

The Hilbert series of a polarised algebraic variety $(X,D)$ is a powerful invariant that, while it captures some features of the geometry of $(X,D)$ precisely, often cannot recover much information about its singular locus. This work…

Following an idea of Ishida, we develop polynomial equations for certain unramified double covers of surfaces with p_g=q=1 and K^2=2. Our first main result provides an explicit surface surface X with these invariants defined over Q that has…

In these notes, we consider self-maps of degree > 1 on a weak del Pezzo surface X of degree < 8. We show that there are exactly 12 such X, modulo isomorphism. In particular, K_X^2 > 2, and if X has one self-map of degree > 1 then for every…

We prove Manin's conjecture for split smooth quintic del Pezzo surfaces over arbitrary number fields with respect to fairly general anticanonical height functions. After passing to universal torsors, we first show that we may restrict the…

This work explores the space of foliations on projective spaces over algebraically closed fields of positive characteristic, with a particular focus on the codimension one case. It describes how the irreducible components of these spaces…

Principal value integrals are associated to multi-valued rational differential forms with normal crossings support on a non-singular algebraic variety. We prove their vanishing on rational surfaces in the context of a conjecture of…