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SAGBI bases with applications to blow-up algebras. (English) Zbl 0866.13010
The authors study the blow-up algebras of certain rational normal scrolls using the theory of SAGBI-bases introduced by L. Robbiano and M. Sweedler [in: Commutative Algebra, Proc. Workshop Salvador 1988, Lect. Notes Math. 1430, 61-87 (1990; Zbl 0725.13013)]. The first sections of the paper are devoted to establish some general results on SAGBI-bases. In particular, the authors show that, if $$A$$ is a subalgebra of the polynomial ring $$R:= K[x_1,\dots,x_n]$$ and if $$\tau$$ is a term-order on $$R$$, then $$\text{in}_\tau(A)$$ (i.e. the $$K$$-algebra generated by the monomials $$\text{in}_\tau(f)$$, $$f\in A$$), when finitely generated, may be viewed as the associated graded ring of a suitable degree filtration on $$A$$. From this result it follows that $$\text{in}_\tau(A)$$ is the special fiber of a flat 1-parameter family whose general fiber is $$A$$. As a consequence, to determine if $$A$$ is normal, Cohen-Macaulay, has rational singularities, or is $$F$$-rational, it is sufficient to check if the corresponding condition is satisfied by $$\text{in}_\tau(A)$$.
If $$I:= (a_1,\dots ,a_m)\subseteq R$$ is an ideal, the Rees algebra $${\mathcal R}(I)= R[It]$$ is a $$K$$-algebra of $$R[t]$$ and if $$a_1,\dots, a_m$$ are homogeneous of the same degree, then the special fiber of $${\mathcal R}(I)$$ may be identified with the $$K$$-subalgebra $$K[a_1,\dots, a_m]$$ of $$R$$. Hence also in this setting it is possible to apply SAGBI bases theory. As a consequence, the authors are able to prove that in some cases, the involved algebras are Cohen-Macaulay or normal and they are able to decide whether $$I$$ is of linear type.
Successively, they consider the class of ideals $$I(c,d)$$ generated by the two minors of a matrix of the form $\begin{pmatrix} x_1 &\ldots &x_i &\ldots &x_c\\ x_{d+1} &\ldots &x_{d+i} &\ldots &x_{c+d}\end{pmatrix}$ which define certain rational normal scrolls. They prove that the natural generators of the Rees algebra and special fiber of $$I(c,d)$$ form a SAGBI-basis, they compute the relations of these algebras and show that they are Cohen-Macaulay normal domains and have rational singularities. Moreover, they are able to prove that the analytic spread of the defining ideal of any rational normal curve in $$\mathbb{P}^n$$ is $$n+1$$, whenever $$n\geq 4$$.
Finally, they study numerical data of the special fiber of $${\mathcal R}(I)$$ (where $$I=I(c,d)$$, as above). They show that the initial ideal of the defining ideal of the special fiber is an ideal of square-free monomials (for a suitable term-order). They show that the attached simplicial complex is shellable and that the Hilbert function is obtained by determining the $$h$$-vectors of the algebra. In particular, this allows to determine those fibers which are Gorenstein.
Reviewer: A.Logar (Trieste)

##### MSC:
 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13C14 Cohen-Macaulay modules 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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