The Gale transform and multi-graded determinantal schemes
Let \(U_i\) be a k-vector space of dimension \( s_i +1 \geq 2\) for \( i = 1, \ldots, k\) and \( k \geq 2\) and \(F\) a vector space of dimension \( n= \sum s_i\) and \( \varphi: F \rightarrow \otimes _{i=1}^k U_i \) a linear transformation and denote by \( \varphi_j: F \otimes k[ \bigoplus_{ i \neq j} U_i ](1,1,1,\dots 1) \) be the corresponding map of free modules. Let \(P_j = \prod_{i \neq j} P^{s_i}\) be the subscheme defined by the ideals of \((s_j +1) \times (s_j +1) \) minor determinants \( I_{s_j +1}( \varphi_j) \subset k[ \bigoplus_{i \neq j}U_i ]\). For any \((k-1)\)-tuple of positive integers \(a\) denote by \( \sigma_a( P_j)\) the Segre embedding given by divisors of type \(a\). The author's main theorem, proposition 3.1, states and proves the equivalence of the following conditions: (i) for all \( 1 \leq i \leq k\), \(\Gamma_i\) is \(0\)-dimensional and locally Gorenstein, \(\text{codim} (I_{s_i +1}( \varphi)) = n-s_i \) and \(\text{codim}( I_{s_i}( \varphi)) = n-s_i +1 \). (ii) \(\Gamma_1\) , \(\Gamma_2\) are \(0\)-dimensional. (iii) \(\text{codim}( I_{s_i + 1}( \varphi)) = n-s_1, \text{codim}( I_{s_1}( \varphi_1)) = n-s_1 + 1 \). (iv) \( \Gamma_1 \) is \(0\)-dimensional and locally Gorenstein. Moreover they prove in theorem 3.2 that if one of the equivalent statements of the above is true then all \(\Gamma_ i \) are isomorphic to each other and defining \( a_i = n -\sum_{l=i}^{k-1}(s_l -1) \) for \(s_k \geq 2 \) and \( b_i = \sum_{l = i+1}^{k-1} s_l \) for \( 1\leq i \leq k-2 \) and \( b_k = n -s_k -1 \). Define \( a= ( a_1, \ldots, a_{k-1}), b= ( b_1, \ldots, b_{k-2}, b_k)\) for \( 1 \leq i \leq k-2\) then the Gale transform of \( \sigma_a( \Gamma_k)\) is \( \sigma_b( \Gamma_{k-1})\). The authors' main motivation is to generalize the results of \textit{D. Eisenbud} and \textit{S. Popescu} [J. Algebra 230, No. 1, 127--173 (2000; Zbl 1060.14528)] to cover linear transformations from a vector space \(F\) to a tensor product of three or more vector spaces, replacing Veronese embeddings by Segre embeddings. The paper is organized as follows. In section one they recall the notion of Gale transform for a linear series stating the result of [loc. cit.] for \(k=2\). In section two for \(\Gamma \) a finite Gorenstein scheme they compute in proposition 2.2 the Gale transform of the linear series cut out as \(\Gamma\) by \( H^0( P, O_P(a))\). In proposition 2.4 by focusing on \(\Gamma_j\) they compute a locally free resolution of \( I_F\) from the Eagon-Northcott complex. For the case of \(\dim(\Gamma_j) = 0 \) the authors compute in proposition 2.6 \(\deg( \Gamma_j)\) using a special case of Porteous formula in proposition 2.5. In section three, they state and prove the main theorem given above and theorem 3.2. The proofs of theorems 3.1 and 3.2 are similar to those of [loc. cit] except for the proof of part b) of theorem 3.2 for which they use proposition 2.4 and proposition 2.6 of this paper.
scientific article; zbMATH DE number 5300799