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Correspondence scrolls. (English) Zbl 1456.14048

This paper introduces schemes called “correspondence scrolls” and gives a general study of them. For a closed subscheme \(Z\) in \(\Pi _{i=1}^n {\mathbb A}^{a_i+1}\), defined by a multigraded ideal \(I\subset A:={\Bbbk}[x_{i,j} : 1\le i \le n, 0 \le j \le a_i]\), and for \(\mathbf{b}=(b_1, \ldots , b_n) \in \mathbb{N}_+^n\), consider the homomorphism \({\Bbbk}[z_{i,\alpha }] \rightarrow A/I\) which sends a variable \(z_{i, \alpha }\) to the monomial \(x_i^\alpha\), of degree \(b_i\). Here \(x_i^ {\alpha }\) denotes \(x_{i,0}^{\alpha _0} \cdots x_{i, a_i}^{\alpha _{a_i}}\). The kernel of the above map defines a closed projective subscheme \(C(Z, {\mathbf b}) \subset { \mathbb P}^N\) (\(N= \sum \binom{a_i+b_i}{a_i}-1)\), called correspondence scroll. This definition includes classical correspondences as well as interesting non-classical ones: rational normal scrolls, double structures which are degenerate \(K3\) surfaces, degenerate Calabi-Yau threefolds, etc. Many invariants or properties of correspondence scrolls are studied: dimension, degree, nonsingularity, Cohen-Macaulay and Gorenstein property and others. The paper is very well written and invites to further research.

MSC:

14J40 \(n\)-folds (\(n>4\))
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13C40 Linkage, complete intersections and determinantal ideals
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14J26 Rational and ruled surfaces
14J28 \(K3\) surfaces and Enriques surfaces
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14M12 Determinantal varieties
14M20 Rational and unirational varieties
14N05 Projective techniques in algebraic geometry
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