Recent zbMATH articles in MSC 14J40 https://www.zbmath.org/atom/cc/14J40 2021-04-16T16:22:00+00:00 Werkzeug Correspondence scrolls. https://www.zbmath.org/1456.14048 2021-04-16T16:22:00+00:00 "Eisenbud, David" https://www.zbmath.org/authors/?q=ai:eisenbud.david "Sammartano, Alessio" https://www.zbmath.org/authors/?q=ai:sammartano.alessio 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 \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/1456.14013 2021-04-16T16:22:00+00:00 "Rizzi, Luca" https://www.zbmath.org/authors/?q=ai:rizzi.luca "Zucconi, Francesco" https://www.zbmath.org/authors/?q=ai:zucconi.francesco Summary: 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.