Related papers: Del Pezzo foliations with log canonical singularit…

We introduce a class of noncommutative surfaces called Artin--Schelter surfaces of del Pezzo types, which contains del Pezzo surfaces as special cases. We show that the moduli stacks of marked Artin--Schelter surfaces of del Pezzo types are…

In this paper, we prove that if a $3$-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>3$ has only log canonical singularities, then so does a general hyperplane section $H$ of $X$. We also…

We consider relatively minimal fibrations of curves of genus two on rational surfaces whose Picard numbers are not maximal. By birational morphisms, such fibred surfaces are interpreted as pencils of plane curves. We show that only four are…

In this paper, we classify Du Val del Pezzo surfaces of Picard rank one in characteristic two and three. We also show that if a Du Val del Pezzo surface is Frobenius split, then a general anti-canonical member is smooth. Furthermore, in…

We consider singular holomorphic foliations on compact complex surfaces with invariant rational nodal curve of positive self-intersection. Then, under some assumptions, we list all possible foliations.

We investigate a geometric criterion for a smooth curve $C$ of genus $14$ and degree $18$ to be described as the zero locus of sections in an Ulrich bundle of rank $3$ on a del Pezzo threefold $V_5 \subset \mathbb{P}^6$. The main challenge…

It is known that the set of log canonical thresholds (lcts) on any varieties with fixed dimension satisfies the ascending chain condition. Inspired by the foliated minimal model program, it is intriguing to study the foliated version of…

We prove the irreducibility of the spaces of rational curves on del Pezzo manifolds of Picard rank 1 and dimension at least 4 by analyzing the fibers of evaluation maps. As a corollary, we prove Geometric Manin's Conjecture in these cases.

We show that smooth quintic del Pezzo threefolds over arbitrary base schemes are classified by non-degenerate ternary symmetric bilinear forms. Then we describe the automorphism group schemes, the Hilbert schemes of lines and the orbit…

The "canonical dimension" of an algebraic group over a field by definition is the maximum of the canonical dimensions of principal homogenous spaces under that group. Over a field of characteristic zero, we prove that the canonical…

In this paper we consider the classification of singularities (Du Val) and real structures (Wall) of weak Del Pezzo surfaces from an algorithmic point of view. It is well known that the singularities of weak Del Pezzo surfaces correspond to…

We give a complete answer for the existence of Kawamata type semiorthogonal decompositions of derived categories of nodal del Pezzo threefolds. More precisely, we show that nodal del Pezzo threefolds of degree $1\leq d \leq 4$ have no…

Third-order symmetric Lorentzian manifolds, i.e. Lorentzian manifold with zero third derivative of the curvature tensor, are classified. These manifolds are exhausted by a special type of pp-waves, they generalize Cahen-Wallach spaces and…

In this paper we classify codimension 1 Mukai foliations on complex projective manifolds

Let X be a K3 surface with an involution g which has non-empty fixed locus X^g and acts non-trivially on a non-zero holomorphic 2-form. We shall construct all such pairs (X, g) in a canonical way, from some better known double coverings of…

We introduce the notion of strongly asymptotically log del Pezzo flags, and classify such flags under the assumption that their zero-dimensional part lies in the boundary. We use this result to give a new and conceptual proof of the…

We classify all complex surfaces with quotient singularities that do not contain any smooth rational curves, under the assumption that the canonical divisor of the surface is not pseudo-effective. As a corollary we show that if $X$ is a log…

This paper reviews Lacini's classification [Lac24] of log del Pezzo surfaces of rank one in characteristics different from two and three, with a focus on where and how Lacini enhanced the techniques of Keel and McKernan [KM99]. We point out…

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. If X contains a prime divisor D with rho(X)-rho(D)>2, then either X is a product of del Pezzo surfaces, or rho(X)=5 or 6. In this setting, we completely classify the case…

We construct motivic cohomology cycles in the group $H^3_{\mathcal M}(Z,{\mathbb Q}(2))$ where $Z$ is a K3 surface obtained as a double cover of a del Pezzo surface $X$ branched at a curve in $|-2K_X|$. The construction uses (-1) curves on…