The Dedekind different of a Cayley–Bacharach scheme

Abstract

Given a 0-dimensional scheme X in a projective space P n K over a field K, we characterize the Cayley-Bacharach property of X in terms of the algebraic structure of the Dedekind different of its homogeneous coordinate ring. Moreover, we characterize Cayley-Bacharach schemes by Dedekind's formula for the conductor and the complementary module, we study schemes with minimal Dedekind different using the trace of the complementary module, and we prove various results about almost Gorenstein and nearly Gorenstein schemes.

An Application of Liaison Theory to Zero-dimensional Schemes

“…In the special case that X is a locally Gorenstein Cayley-Bacharach scheme, the regularity index of the Dedekind different attains the maximal value. This also follows from [14,Proposition 4.8] with a different proof. Corollary 4.9.…”

“…Later investigations of the Cayley-Bacharach property have included the work of Fouli, Polini, and Ulrich [6], Robbiano [17], Gold, Little, and Schenck [8], and Guardo [10]. Moreover, this property has also been extended for 0-dimensional schemes in P n K (see [14][15][16]21]). When X ⊆ P n K is a 0-dimensional scheme over an algebraically closed field K, Robbiano and the first author [17] considered subschemes of X of degree deg(X) − 1 to show that the conditions (a) and (c) of Theorem 1.1 are still equivalent.…”

“…Our focus in this paper is to look at an extension of the Cayley-Bacharach property and to generalize the above theorem for 0-dimensional schemes X in P n K over an arbitrary field K. In particular, we will look closely at the natural question whether conditions (a) and (b) of the above theorem are equivalent for our more general setting. Our approach is to use the notion of maximal p j -subschemes of X which are introduced and studied in the papers [13,14]. Also, we discuss a characterization of the Cayley-Bacharach property of degree d with d ∈ N in terms of the canonical module of the coordinate ring of X and apply this result to bound the Hilbert function of the Dedekind different of X and determine its regularity index in some special cases.…”

Nearly Gorenstein cyclic quotient singularities

“…On the other hand, nearly Gorenstein rings, which are the object of interest of this paper, have been introduced even more recently by Herzog et al (2019) in 2019, although their defining property was already examined by Ding (1993), Huneke and Vraciu (2006), and Striuli and Vraciu (2011). Moreover, nearly Gorenstein rings have been studied in several contexts, such as zero-dimensional schemes (Kreuzer et al 2019), affine semigroup rings Herzog et al (2019), and affine monomial curves Moscariello (2020). See also Endo et al (2020), Dao et al (2020), Kobayashi (2020), Kumashiro (2020), Rahimi (2020) for other related results.…”

“…On the other hand, nearly Gorenstein rings, which are the object of interest of this paper, have been introduced even more recently by Herzog, Hibi, and Stamate [HHS19] in 2019. Nevertheless, they have already been studied in several contexts, such as zero-dimensional schemes [KLL19], affine semigroup rings [HJS19], and affine monomial curves [MS20]. See also [EGI19,DKT20,Kob20,Rah20] for other related results.…”