Seshadri constants on principally polarized abelian surfaces with real multiplication

Seshadri constants on abelian surfaces are fully understood in the case of Picard number one. Little is known so far for simple abelian surfaces of higher Picard number. In this paper we investigate principally polarized abelian surfaces with real multiplication. They are of Picard number two and might be considered the next natural case to be studied. The challenge is to not only determine the Seshadri constants of individual line bundles, but to understand the whole Seshadri function on these surfaces. Our results show on the one hand that this function is surprisingly complex: on surfaces with real multiplication in $$\mathbb {Z}[\sqrt{e}]$$ Z [ e ] it consists of linear segments that are never adjacent to each other—it behaves like the Cantor function. On the other hand, we prove that the Seshadri function is invariant under an infinite group of automorphisms, which shows that it does have interesting regular behavior globally.

Seshadri constants on some Quot schemes

Let E be a vector bundle of rank n on ℙ 1 {\mathbb{P}^{1}} . Fix a positive integer d. Let 𝒬 ⁢ ( E , d ) {\mathcal{Q}(E,d)} denote the Quot scheme of torsion quotients of E of degree d and let Gr ⁢ ( E , d ) {\mathrm{Gr}(E,d)} denote the Grassmann bundle that parametrizes the d-dimensional quotients of the fibers of E. We compute Seshadri constants of ample line bundles on 𝒬 ⁢ ( E , d ) {\mathcal{Q}(E,d)} and Gr ⁢ ( E , d ) {\mathrm{Gr}(E,d)} .

Applications of a Grassmannian technique to hyperbolicity, Chow equivalency, and Seshadri constants

In this paper we further develop a Grassmannian technique used to prove results about very general hypersurfaces and present three applications. First, we provide a short proof of the Kobayashi conjecture given previously established results on the Green–Griffiths–Lang conjecture. Second, we completely resolve a conjecture of Chen, Lewis, and Sheng on the dimension of the space of Chow-equivalent points on a very general hypersurface, proving the remaining cases and providing a short, alternate proof for many of the previously known cases. Finally, we relate Seshadri constants of very general points to Seshadri constants of arbitrary points of very general hypersurfaces.

Integrality of Seshadri constants and irreducibility of principal polarizations on products of two isogenous elliptic curves

In this paper we consider the question of when all Seshadri constants on a product of two isogenous elliptic curves $$E_1\times E_2$$ E 1 × E 2 without complex multiplication are integers. By studying elliptic curves on $$E_1\times E_2$$ E 1 × E 2 we translate this question into a purely numerical problem expressed by quadratic forms. By solving that problem, we show that all Seshadri constants on $$E_1\times E_2$$ E 1 × E 2 are integers if and only if the minimal degree of an isogeny $$E_1\rightarrow E_2$$ E 1 → E 2 equals 1 or 2. Furthermore, this method enables a characterization of irreducible principal polarizations on $$E_1\times E_2$$ E 1 × E 2 .

Simple cyclic covers of the plane and the Seshadri constants of some general hypersurfaces in weighted projective space

Let $$X \subset {\mathbb P}(1,1,1,m)$$ X ⊂ P ( 1 , 1 , 1 , m ) be a general hypersurface of degree md for some for $$d\ge 2$$ d ≥ 2 and $$m\ge 3$$ m ≥ 3 . We prove that the Seshadri constant $$\varepsilon ( {\mathcal O}_X(1), x)$$ ε ( O X ( 1 ) , x ) at a general point $$x\in X$$ x ∈ X lies in the interval $$\left[ \sqrt{d}- \frac{d}{m}, \sqrt{d}\right] $$ d - d m , d and thus approaches the possibly irrational number $$\sqrt{d}$$ d as m grows. The main step is a detailed study of the case where X is a simple cyclic cover of the plane.