Correspondence Scrolls
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Correspondence Scrolls David Eisenbud1
· Alessio Sammartano2
Received: 16 June 2018 / Revised: 23 August 2018 / Accepted: 7 September 2018 / Published online: 2 October 2018 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018
Abstract This paper initiates the study of a class of schemes that we call correspondence scrolls, which includes the rational normal scrolls and linearly embedded projective bundle of decomposable bundles, as well as degenerate K3 surfaces, Calabi-Yau threefolds, and many other examples. Keywords Rational normal scroll · Veronese embedding · Join variety · Multiprojective space · Variety of complexes · Variety of minimal degree · Double structure · K3 surface · Calabi-Yau scheme · Gorenstein ring · Gr¨obner basis Mathematics Subject Classification (2010) Primary 14J40; Secondary 13H10 · 13C40 · 13P10 · 14J26 · 14J28 · 14J32 · 14M05 · 14M12 · 14M20
1 Introduction We will define and study a class of schemes that we call correspondence scrolls. The origin of our interest was in a paper by Frank Schreyer and the first author on the equations and syzygies of degenerate K3 surfaces such as K3 carpets [13]. Correspondence scrolls are a natural generalization of rational normal scrolls and K3 carpets that includes families of (degenerate) Calabi-Yau threefolds and many other examples. We will define a correspondence scroll C(Z; b) for any subscheme Z ⊆ Pa := Pa1 × · · · × Pan , and any n-tuple of non-negative integers b = (b1 , . . . , bn ). In the special case where Z is reduced, we may define C(Z; b) as follows: embed Pai by the bi -th Veronese David Eisenbud
[email protected] Alessio Sammartano [email protected] 1
Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA, 94720, USA
2
Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN, 46556, USA
102
D. Eisenbud, A. Sammartano
Pai
embedding → P a + b νbi : − 1, and set a i i
ai + bi ai
−1
into a general linear subspace of PN , where N =
i
i
C(Z; b) =
νb1 (p1 ), . . . , νbn (pn ).
(p1 ,...,pn )∈Z
Thus, C(Z; b) is a union of (n − 1)-planes. In this paper, we determine the dimension, degree, and multigraded Hilbert function of a scheme of the form C(Z; b), and for which Z they are nonsingular for every b. We explain the primary decompositions and Gr¨obner bases of their defining ideals, and we determine which ones are Cohen-Macaulay, Gorenstein, or numerically Calabi-Yau. We give numerous examples, including some new (as far as we know) examples of degenerate Calabi-Yau threefolds. Recall that rational normal scrolls are the varieties of minimal degree in PN that contain linear spaces of codimension 1 (the only other varieties of minimal degree are the cones over the Veronese surface in P5 , see [12] for an expository account). Perhaps because of their extremal properties, they appear in many contexts in algebraic geometry, for example, as ambient spaces of Castlenuovo curves (see, for example, [17]) and canonical cu
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