Recent zbMATH articles in MSC 14J40https://www.zbmath.org/atom/cc/14J402021-04-16T16:22:00+00:00WerkzeugCorrespondence scrolls.https://www.zbmath.org/1456.140482021-04-16T16:22:00+00:00"Eisenbud, David"https://www.zbmath.org/authors/?q=ai:eisenbud.david"Sammartano, Alessio"https://www.zbmath.org/authors/?q=ai:sammartano.alessioThis 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 \textit{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.
Reviewer: Nicolae Manolache (Bucureşti)Differential forms and quadrics of the canonical image.https://www.zbmath.org/1456.140132021-04-16T16:22:00+00:00"Rizzi, Luca"https://www.zbmath.org/authors/?q=ai:rizzi.luca"Zucconi, Francesco"https://www.zbmath.org/authors/?q=ai:zucconi.francescoSummary: We extend the theory of \textit{G. P. Pirola} and \textit{F. Zucconi} [J. Algebr. Geom. 12, No. 3, 535--572 (2003; Zbl 1083.14515)]. We introduce the new notion of adjoint quadric for canonical images of irregular varieties. Using this new notion, we obtain the infinitesimal Torelli theorem for varieties whose canonical image is a complete intersection of hypersurfaces of degree \(>2\) and for Schoen surfaces. Finally, we show that a family with fiberwise liftable holomorphic forms such that the fibers have Albanese morphism of degree 1 is birationally trivial if there exist no adjoint quadrics.