## 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.

### MSC:

 14M12 Determinantal varieties 13C40 Linkage, complete intersections and determinantal ideals 13C15 Dimension theory, depth, related commutative rings (catenary, etc.)

### Keywords:

ideal of minors; Artinian Gorenstein algebras

### Citations:

Zbl 0976.14031; Zbl 0927.13020
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