Basepoint-freeness thresholds and higher syzygies on abelian threefolds

Abstract

For a polarized abelian variety, Z. Jiang and G. Pareschi introduce an invariant and show that the polarization is basepoint free or projectively normal if the invariant is small. Their result is generalized to higher syzygies by F. Caucci, that is, the polarization satisfies property (Np) if the invariant is small.In this paper, we study a relation between the invariant and degrees of abelian subvarieties with respect to the polarization. For abelian threefolds, we give an upper bound of the invariant using degrees of abelian subvarieties. In particular, we affirmatively answer a question about (Np) on abelian varieties asked by the author and V. Lozovanu in the three dimensional case.

Higher syzygies on general polarized abelian varieties of type $(1,\dots,1,d)$

“…, 1, d) and dimension g with d g i=0 (p + 2) i . In [Ito20], the author observes a similarity between β(l) −1 and Seshadri constants. Since Seshadri constants have a lower-semicontinuity, it is natural to ask whether β(l) is uppersemicontinuous or not.…”

Cohomological rank functions and Syzygies of abelian varieties

“…Let D = 1 p+2 L. Since (D g ) > (g) g , we may assume that D is an effective Q-divisor with m o (D) > g. Then there exists D 1 = c 1 D for some 0 < c 1 < 1 such that (A, D 1 ) is log canonical at o and the minimal lc center through o is an normal subvariety Z 1 of dimension k ≥ 0. If k = 0, we know that r ′ (L) = 1 p+2 r ′ (D) < 1 p+2 and we conclude by [Ito2,Proposition 1.10].…”

“…Ito ([Ito1] and [Ito2]) proved Conjecture 1.4 for abelian surfaces and abelian threefolds. He also proved that ([Ito2,Proposition 3.3…”

Gaussian Gabor frames, Seshadri constants and generalized Buser--Sarnak invariants

“…) follows directly from the Balian-Low type theorem. To prove (3), we shall use the following inequality (see [30,Lemma 3…”