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Given a 0-dimensional scheme $mathbb{X}$ in a projective space $mathbb{P}^n_K$ over a field $K$, we characterize the Cayley-Bacharach property of $mathbb{X}$ in terms of the algebraic structure of the Dedekind different of its homogeneous coordinate ring. Moreover, we characterize Cayley-Bacharach schemes by Dedekinds 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.
In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley-Bacharach condition. In particular, by bounding the number of points satisfying the Cayley-Bacharach condition, we force them to lie on uni
The Cayley-Bacharach property, which has been classically stated as a property of a finite set of points in an affine or projective space, is extended to arbitrary 0-dimensional affine algebras over arbitrary base fields. We present characterizations
We introduce a theory of multigraded Cayley-Chow forms associated to subvarieties of products of projective spaces. Two new phenomena arise: first, the construction turns out to require certain inequalities on the dimensions of projections; and secon
We introduce a new class of $mathfrak{sl}_2$-triples in a complex simple Lie algebra $mathfrak{g}$, which we call magical. Such an $mathfrak{sl}_2$-triple canonically defines a real form and various decompositions of $mathfrak{g}$. Using this decompo
We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A_n singularities. The operators encoding these invariants are expressed in terms of the action of t