SAGBI bases with applications to blow-up algebras.

*(English)*Zbl 0866.13010The 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.

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) |