A note on the cone of mobile curves

A note on the Kuznetsov component of the Veronese double cone

This note describes moduli spaces of complexes in the derived category of a Veronese double cone $Y$. Focusing on objects with the same class $κ_1$ as ideal sheaves of lines, we describe the moduli space of Gieseker stable sheaves and show that it has two components. Then, we study the moduli space of stable complexes in the Kuznetsov component of $Y$ of the same class, which also has two components. One parametrizes ideal sheaves of lines and it appears in both moduli spaces. The other components are not directly related by a wall-crossing: we show this by describing an intermediate moduli sp…

Nef Cone of a Generalized Kummer 4-fold

In this note, we calculate the boundary of movable cones and nef cones of the generalized Kummer 4-fold $\mathrm{Km}^2(A)$ attached to an abelian surface $A$ with $\mathrm{rkNS}(A) = 1$.

Pulling subdivisions of cones and blowups of monomial ideals on affine toric varieties

This short note solves the following problem: Given a map of normal toric varieties corresponding to a coherent subdivision of a cone, find an ideal such that the given map is the blowup of that ideal.

A note on the cone conjecture for K3 surfaces in positive characteristic

We prove that, for a K3 surface in characteristic p > 2, the automorphism group acts on the nef cone with a rational polyhedral fundamental domain and on the nodal classes with finitely many orbits. As a consequence, for any non-negative integer g, there are only finitely many linear systems of irreducible curves on the surface of arithmetic genus g, up to the action of the automorphism group.

Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone

The purpose of this note is to study the number of elements in Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone.

The cone of moving curves of a smooth Fano-threefold

In this short note we show that the closed cone of moving curves of a smooth Fano-threefold is polyhedral. The proof relies on a famous result of Bucksom, Demailly, Paun and Peternell which says that the strongly movable cone is dual to the cone of pseudoeffective divisors. Finally, we relate the extremal rays of the cone of moving curves on a threefold with extremal rays in the cone of strongly movable curves on a birational model obtained by a Mori contraction.

A note on non-commutative polytopes and polyhedra

It is well-known that every polyhedral cone is finitely generated (i.e. polytopal), and vice versa. Surprisingly, the two notions differ almost always for non-commutative versions of such cones. This was obtained as a byproduct in an earlier paper. In this note we give a direct and constructive proof of the statement. Our proof also yields a surprising quantitative result: the difference of the two notions can always be seen at the first level of non-commutativity, i.e. for matrices of size $2$, independent of dimension and complexity of the initial convex cone.

Effective cone of the blow up of the symmetric product of a curve

Let $C$ be a smooth curve of genus $g \geq 1$ and let $C^{(2)}$ be its second symmetric product. In this note we prove that if $C$ is very general, then the blow-up of $C^{(2)}$ at a very general point has non-polyhedral pseudo-effective cone. The strategy is to consider first the case of hyperelliptic curves and then to show that having polyhedral pseudo-effective cone is a closed property for families of surfaces.

关于多面体锥和环面多对数的一个注记

A note on polyhedral cones and toric polylogarithms

本文通过建立单纯锥链复形与Milnor K-理论Gersten复形之间的同构，推广了先前关于代数环面上Milnor K-理论特殊元素的方法，改进了Charlton-Radchenko-Rudenko的结果。

The cone of pseudo-effective divisors of log varieties after Batyrev

In these notes we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective $\mathbb Q$-factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev's structure theorem for the cone of nef curves of projective terminal threefolds.

Nilpotent cones and adjoint orbits

A short note to show that the elements of the (open) cone underlying a nilpotent orbit on a period domain are pairwise congruent under the symmetry group of the period domain.

五次圆锥形奇点的小消解的平衡锥

The balanced cone of the small resolution of the quintic conifold

本文通过相交数方法，显式确定了五次圆锥形奇点的小消解的平衡锥，为相关几何结构提供了精确刻画。

Partially ample line bundles on toric varieties

In this note we study properties of partially ample line bundles on simplicial projective toric varieties. We prove that the cone of q-ample line bundles is a union of rational polyhedral cones, and calculate these cones in examples. We prove a restriction theorem for big q-ample line bundles, and deduce that q-ampleness of the anticanonical bundle is not invariant under flips. Finally we prove a Kodaira-type vanishing theorem for q-ample line bundles.

关于Enriques流形的锥猜想

On the cone conjecture for Enriques manifolds

本文证明当覆盖度为素数时，极一般Enriques流形满足Morrison-Kawamata锥猜想，并验证了与已知例子形变等价的极一般Enriques流形也满足该猜想。

Compact subvarieties with ample normal bundles, algebraicity and cones of cycles

In this note we study two features of submanifolds (subvarieties) with ample normal bundles in a compact Kähler manifold X. First, we study how algebraic X can be, i.e. we investigate the algebraic dimension. Second, we study curves with ample normal bundles in case X is projective. We prove in various cases that the class of the curve is in the interior of the Mori cone.

关于有界比率的注记

A note on bounded ratios

该数学注记证明半代数集上的有界比率集合为有理多面体凸锥，给出洛伦兹多项式有界比率相关猜想的反例，关联7顶点割锥的非超度量团-网面。