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Cohomological rank functions and syzygies of abelian varieties. (English) Zbl 1502.14101

Let \(X\) be a smooth projective variety, and take \(L\) a very ample line bundle on \(X\). A very natural question one can ask is how “nice” is the projective embedding induced by \(L\). What is the maximum degree of the equations defining \(X\)? What is the degree of the relations between these equations? And the degree of the relations between the relations? We say that the polarized variety \((X,L)\) satisfies property \((N_p)\) if these degree are as small as possible for the first \(p\) steps of the graded free minimal resolutions of the algebra \(R_L:=\bigoplus_mH^0(X,L^{\otimes m})\) over the polynomial ring \(S_L:=\operatorname{Sym}H^0(X,L)\). In particular \(N_0\) means that the embedding is projectively normal, \(N_1\) means that \(X\) is defined by quadric equations, \(N_2\) is equivalent that the relations between the quadric equation of \(X\) are linear and so on.
This paper is concerned with the study of property \((N_p)\) when \(X\) is an abelian variety and \(L\) is a primitive polarization, that is when \(L\) is not a power of an ample line bundle.
The main tool is the cohomological rank function – introduced in [Z. Jiang and G. Pareschi, Ann. Sci. Éc. Norm. Supér. (4) 53, No. 4, 815–846 (2020; Zbl 1459.14006)] by elaborating on ideas developed in [M. Á. Barja et al., J. Inst. Math. Jussieu 19, No. 6, 2087–2125 (2020; Zbl 1452.14005)] – which was first applied to the study of syzygies in [F. Caucci, Algebra Number Theory 14, No. 4, 947–960 (2020; Zbl 1442.14140)]. The main results of these paper are progresses toward a conjecture of A. Ito [Commun. Algebra 46, No. 12, 5342–5347 (2018; Zbl 1411.14016)] and V. Lozovanu [“Singular divisors and syzygies of polarized abelian threefolds”. Preprint, arXiv:1803.08780], and the proof of a conjectur of Caucci [loc. cit.] in the case of a very general polarized abelian variety.

MSC:

14K05 Algebraic theory of abelian varieties
14F17 Vanishing theorems in algebraic geometry
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
13D02 Syzygies, resolutions, complexes and commutative rings
14K12 Subvarieties of abelian varieties
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