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On syzygies of non-complete embedding of projective varieties. (English) Zbl 1137.13008

Motivated from properties \(N_p\) and \(N_{d,p}\) first studied by Green and Lazarsfeld, the last two authors defined property \(N^S_p\) for non-complete embedding of projective varieties [S. Kwak and E. Park, J. Reine Angew. Math. 582, 87–105 (2005; Zbl 1076.14064)]. Roughly speaking, if \(X\) is a projective variety, \(W\) is a very ample subsystem of \(H^0(X, O_X(1))\) and \(S\) is the homogeneous coordinate ring of \(\mathbb{P}(W)\), then saying that \(X\) satisfies property \(N^S_p\) if the coordinate ring of \(X\) viewed as an \(S\)-module is generated in degree 0 and 1, and has linear sysygies up to the \(p\)-th step.
In the paper under review, the authors continue the study in [loc. cit.], examining algebraic and geometric interpretations of property \(N^S_p\). Their main results include the following theorems, which extend results in [loc. cit.].
Theorem 2. Let \(X \subseteq \mathbb{P}^r\) be a reduced non-degenerate projective variety which satisfies property \(N^S_p\). If \(X \subseteq \mathbb{P}^{r-t}\) is an isomorphic linear projection, where \(0 \leq t \leq p\), then \(X \subseteq \mathbb{P}^r\) satisfies property \(N^S_{p-t}\).
Theorem 3. Let \(X \subseteq \mathbb{P}^r\) be a reduced non-degenerate projective variety. Assume that
(i) property \(N^S_p\) holds for some \(p \geq 1\), and
(ii) \(k\)-normality holds for all \(k \geq k_0\).
Then \(X\) satisfies property \(N_{k_0+1,p}\).
The authors also study hyperplane section of a variety that satisfies property \(N^S_p\). Roughly speaking, it is shown that if a variety satisfies property \(N^S_p\) then so does a general hyperplane section (see Theorem 4).

MSC:

13D02 Syzygies, resolutions, complexes and commutative rings
14N15 Classical problems, Schubert calculus

Citations:

Zbl 1076.14064
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References:

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