Linear syzygies of curves in weighted projective space

Abstract

We develop analogues of Green's N p -conditions for subvarieties of weighted projective space, and we prove that such N p -conditions are satisfied for high degree embeddings of curves in weighted projective space. A key technical result links positivity with low degree (virtual) syzygies in wide generality, including cases where normal generation fails.

Positivity and nonstandard graded Betti numbers

“…In the classical setting, Corollary 1.8 implies that 𝑀 ⩾𝑟 (𝑟) has a linear resolution, as 𝑤 𝑖 = 𝑖 = 𝑤 𝑖 for all 𝑖. Thus, the conditions in this corollary can be seen as providing a nonstandard graded analogue of a linear resolution; in fact, this notion of a "Koszul linear" complex arises in [7] in relation to 𝑁 𝑝 -conditions on weighted projective space, and it contrasts with the notion of strong linearity from [5, Definition 1.2].…”

“…In fact, a main theme from recent work on syzygies with nonstandard gradings, for example, [3, 5, 7, 9-11, 14, 18, 20, 23], is that notions from the standard graded case can have several distinct nonstandard graded analogues, each of which is useful for different purposes. For instance, several analogs of linear resolutions in the nonstandard ℤ-graded setting play a key role in [7]. Our goal in this paper is to demonstrate how an alternate analogue of regularity -Koszul regularity -can provide sharper information in some contexts.…”

“…Weighted regularity has had significant applications, for instance, to group cohomology and invariant theory [8,25,26] as well as to our work on 𝑁 𝑝 -conditions in weighted projective spaces [7]. To define Koszul regularity, we need the following notation.…”

“…The degrees of the generators of the truncation 𝑆 ⩾𝑟 (𝑟) depend on the remainder of 𝑟 divided by 10. For instance, 𝑆 ⩾1 ( 1) is generated by 𝑥 and 𝑦 in degrees 0 and 9, whereas 𝑆 ⩾7 (7) is generated by 𝑥 7 and 𝑦 in degree 0 and 3. Thus, the generating degrees of 𝑆 ⩾𝑟 (𝑟) -that is, the Betti numbers in homological degree 0 -depend on the maximal degree of a variable, that is, 𝑤 1 .…”