Local positivity, multiplier ideals, and syzygies of abelian varieties

Abstract

We use the language of multiplier ideals in order to relate the syzygies of an abelian variety in a suitable embedding with the local positivity of the line bundle inducing that embedding. This extends to higher syzygies a result of Hwang and To on projective normality.

Geometric aspects of Newton–Okounkov bodies

“…Our approach builds in part on the method of proof developed in [LPP10]. For the record we state the final result of [LPP10]. Theorem 3.3.1 (Lazarsfeld-Pareschi-Popa).…”

“…Show that L is very ample, but not projectively normal. The essential novelty of our proof is the use of infinitesimal Newton-Okounkov polygons to construct effective Q-divisors whose multiplier ideal coincides with the maximal ideal of the origin; this replaces the straightforward genericity argument of [LPP10].…”

“…From a technical point of view, the essential contribution of the work [LPP10] can be summarized in the following statement. For the sake of clarity we give a quick outline of the argument in [LPP10]; to this end, we quickly recall some terminology. As above, p will denote a natural number, one studies sheaves on the (p + 2)-fold self-product X ×(p+2) .…”

“…To ease the presentation we assume that there exists a Seshadri-exceptional curve F ⊆ X passing through the origin o ∈ X with the property that r def = (L · F) q = mult o (F) 2 and ε def = ε(L; o) = r/q. Our starting point is the method of [LPP10], which builds on the following observation of Green [Gre84] (see also [Ina97]): consider the diagonal ∆ ⊆ X × X with ideal sheaf I ∆ . Projective normality of L is equivalent to the vanishing condition (3.13.6)…”

“…The authors of [LPP10] then go on to show that in order to guarantee the vanishing in (3.13.6), it suffices to verify the existence of an effective Q-divisor…”