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**The dimension of certain catalecticant varieties.**
*(English)*
Zbl 1072.14064

Consider the lexicographically ordered list of all 3-tuples \(\underline{i}=(i_1,i_2,i_3)\in \mathbb Z_{\geq 0}\) for which \(| i| =i_1+i_2+i_3=t.\) Let \(o(\underline{i})\) be the position of \(\underline{i}\) in the list. Let Cat\((t,t;3)\) be the \({t+2\choose 2}\times {t+2\choose 2}\) matrix whose \(o(\underline{i}),o(\underline{j})\)- entry is \([Y_{\underline{i}+\underline{j}}].\) Hereby \({2t+2\choose 2}\) variables \(Y_{\underline{k}}\) are introduced. In the associated polynomial ring \(k[\{Y_{\underline{k}}: | \underline{k}| =2t \}],\) \(k\) an algebraically closed field, let \(I_{s+1,t}\) be the ideal of \((s+1)\times (s+1)\)-minors of Cat\((t,t;3),\) and let \(V_{s,t}\) be the associated projective variety. The \(V_{s,t}\) are examples of catalecticant varieties, living in projective space \(\mathbb P^N,\) \(N={2t+2 \choose 2}-1.\)

Let \(\tilde{s}={t+2\choose 2}-s.\) Standard results on ideals of minors and in dimension theory of algebraic varieties guarantee that \(\dim V_{s,t}\geq \text{ exdim}V_{s,t}:=\max\{0, N-{\tilde{s}+1 \choose 2} \},\) which latter number authors call the expected dimension of \(V_{s,t}.\) S. Diesel had conjectured that if \(\tilde{s}\leq t+1,\) then \(\dim V_{s,t}=\text{ exdim}V_{s,t}.\) This was shown to be false by Y. H. Cho and B. E. Jung [Commun. Algebra 28, No. 5, 2423–2443 (2000; Zbl 0976.14031)]. Using previous results of theirs [A. Conca and G. Valla, Math. Z. 230, No. 4, 753–784 (1999; Zbl 0927.13020)], the authors give a complete answer to when equality holds as follows:

Let \(\Gamma=(a_0=1,2,\ldots,a_t)\) be a sequence of integers satisfying for some \(m\in \{2,\ldots, t+1\}\) that \(a_i=i+1,\) if \(0\leq i\leq m-1,\) and \(0\leq a_{i+1} \leq a_i,\) if \(m-1\leq i\leq t,\) and \(\sum a_i=s;\) let \(\tilde{V}_{s,t}\) be the set of all such sequences; and put \(\rho(\Gamma)=2s-\sum_{i=0}^{t-2} a_i(a_{i+2}-a_{i+1})+2a_{t-1}a_t - a_t(a_t+3)/2.\) Let \(T=\{1,3,h_2,\dots,h_{2t-2},3,1\}\) be a Gorenstein sequence of socle degree \(2t\) and with \(h_t=s;\) let \(a_i=h_i-h_{i-1}.\) To certain such sequences varieties Gor\((T)\) are associated whose union is essentially \(V_{s,t}.\)

Via Gonca and Valla [loc.cit.] the authors get Proposition 2.1: \(\dim\)Gor\((T)=\rho(\Gamma);\) and from this the crucial result:

Theorem 2.4: If \(t\geq 2\) and \(3 \leq s <{t+2\choose 2},\) then \(\dim V_{s,t}=\max\{\rho(\Gamma): \Gamma\in \tilde{V}_{s,t} \}.\) By a tower the authors understand a sequence of \(t+1\) integers given for \({a+2\choose 2}\leq \tilde{s} \leq t+1\) by \[ \Gamma_a:= \begin{matrix} 0& 1& \dots & t-a-1 & \dots &t-1 & t \\ 1& 2& \dots & t-a & \dots &t-a & t+1-\tilde{s}+{a+1\choose 2} \end{matrix}. \] Such sequences are in \(\tilde{V}_{s,t}.\)

Next, some elementary but subtle lemmas yield

Theorem 3.9: If \(\tilde{s} \leq \min\{t,\frac{2t}{3}+4\},\) then \(\dim V_{s,t} = \max \{\rho(\Gamma_a): \Gamma_a \text{ a tower in } \tilde{V}_{s,t} \}.\)

Defining the rational functions \(D(X)=\frac{1}{4} (X^4+6X^3+11X^2+30X+8)\) and \(f_t(X)=\frac{4Xt+D(X)}{(X+1)(X+2)}\) this leads to

Theorem 3.11. \(V_{s,t}\) has the expected dimension if and only if \(\tilde{s}\leq f_t(a)\) for every integer \(a\geq 1\) for which \({a+2\choose 2}\leq \tilde{s}.\)

This necessary and sufficient criterion is morphed in section 4 into a criterion of the form that \(V_{s,t}\) has the expected dimension if and only if \(\tilde{s}\leq N(t).\) The explanation of \(N(t)\) again would take some lines. Reviewer ventures to say that questions like determining \(N(t)\) departing from theorem 3.11 can – in principle at least – be “automatically” resolved via quantifier elimination.

Let \(\tilde{s}={t+2\choose 2}-s.\) Standard results on ideals of minors and in dimension theory of algebraic varieties guarantee that \(\dim V_{s,t}\geq \text{ exdim}V_{s,t}:=\max\{0, N-{\tilde{s}+1 \choose 2} \},\) which latter number authors call the expected dimension of \(V_{s,t}.\) S. Diesel had conjectured that if \(\tilde{s}\leq t+1,\) then \(\dim V_{s,t}=\text{ exdim}V_{s,t}.\) This was shown to be false by Y. H. Cho and B. E. Jung [Commun. Algebra 28, No. 5, 2423–2443 (2000; Zbl 0976.14031)]. Using previous results of theirs [A. Conca and G. Valla, Math. Z. 230, No. 4, 753–784 (1999; Zbl 0927.13020)], the authors give a complete answer to when equality holds as follows:

Let \(\Gamma=(a_0=1,2,\ldots,a_t)\) be a sequence of integers satisfying for some \(m\in \{2,\ldots, t+1\}\) that \(a_i=i+1,\) if \(0\leq i\leq m-1,\) and \(0\leq a_{i+1} \leq a_i,\) if \(m-1\leq i\leq t,\) and \(\sum a_i=s;\) let \(\tilde{V}_{s,t}\) be the set of all such sequences; and put \(\rho(\Gamma)=2s-\sum_{i=0}^{t-2} a_i(a_{i+2}-a_{i+1})+2a_{t-1}a_t - a_t(a_t+3)/2.\) Let \(T=\{1,3,h_2,\dots,h_{2t-2},3,1\}\) be a Gorenstein sequence of socle degree \(2t\) and with \(h_t=s;\) let \(a_i=h_i-h_{i-1}.\) To certain such sequences varieties Gor\((T)\) are associated whose union is essentially \(V_{s,t}.\)

Via Gonca and Valla [loc.cit.] the authors get Proposition 2.1: \(\dim\)Gor\((T)=\rho(\Gamma);\) and from this the crucial result:

Theorem 2.4: If \(t\geq 2\) and \(3 \leq s <{t+2\choose 2},\) then \(\dim V_{s,t}=\max\{\rho(\Gamma): \Gamma\in \tilde{V}_{s,t} \}.\) By a tower the authors understand a sequence of \(t+1\) integers given for \({a+2\choose 2}\leq \tilde{s} \leq t+1\) by \[ \Gamma_a:= \begin{matrix} 0& 1& \dots & t-a-1 & \dots &t-1 & t \\ 1& 2& \dots & t-a & \dots &t-a & t+1-\tilde{s}+{a+1\choose 2} \end{matrix}. \] Such sequences are in \(\tilde{V}_{s,t}.\)

Next, some elementary but subtle lemmas yield

Theorem 3.9: If \(\tilde{s} \leq \min\{t,\frac{2t}{3}+4\},\) then \(\dim V_{s,t} = \max \{\rho(\Gamma_a): \Gamma_a \text{ a tower in } \tilde{V}_{s,t} \}.\)

Defining the rational functions \(D(X)=\frac{1}{4} (X^4+6X^3+11X^2+30X+8)\) and \(f_t(X)=\frac{4Xt+D(X)}{(X+1)(X+2)}\) this leads to

Theorem 3.11. \(V_{s,t}\) has the expected dimension if and only if \(\tilde{s}\leq f_t(a)\) for every integer \(a\geq 1\) for which \({a+2\choose 2}\leq \tilde{s}.\)

This necessary and sufficient criterion is morphed in section 4 into a criterion of the form that \(V_{s,t}\) has the expected dimension if and only if \(\tilde{s}\leq N(t).\) The explanation of \(N(t)\) again would take some lines. Reviewer ventures to say that questions like determining \(N(t)\) departing from theorem 3.11 can – in principle at least – be “automatically” resolved via quantifier elimination.

Reviewer: Alexander Kovačec (Coimbra)