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On pseudo symmetric monomial curves. (English) Zbl 1412.13027

Monomial curves, toric ideals and monomial algebras associated to 4-generated pseudo symmetric numerical semigroups are studied in this work.
Let \(n_1,n_2,n_3,n_4\) be positive integers with \(\gcd(n_1,n_2,n_3,n_4)=1\), \(S\) be the numerical semigroup generated by \(n_1,\dots,n_4\), and \(K[S]=K[t^{n_1},\dots,t^{n_4}]\) be the semigroup ring of \(S\). Then, \(K[S]\simeq A/I_S\) where \(A=K[X_1,\dots,X_4]\) and the ideal \(I_S\) is the kernel of the surjection \(A\to K[S]\) where \(X_i\to t_{n_i}\). When \(S\) is pseudo-symmetric, then \(K[S]\simeq A/(f_1,f_2,f_3,f_4,f_5)\), where \begin{align*} & f_1=X_1^{\alpha_1}-X_3X_4^{\alpha_4-1}, f_2=X_2^{\alpha_2}-X_1^{\alpha_{21}}X_4 \\ & f_3=X_3^{\alpha_3}-X_1^{\alpha_1-\alpha_{21}}X_2 f_4=X_4^{\alpha_4}-X_1X_2^{\alpha_2-1}X_3^{\alpha_3-1} \\ & f_5=X_3^{\alpha_3-1}X_1^{\alpha_{21}+1}-X_2X_4^{\alpha_4-1}, \end{align*} with \(\alpha_i>1\), \(\alpha_{21}>0\), \(\alpha_{21}<\alpha_1\), \(n_1=\alpha_2\alpha_3(\alpha_4-1)+1\), \(n_2=\alpha_{21}\alpha_3\alpha_4+(\alpha_1-\alpha_{21}-1)(\alpha_3-1)+\alpha_3\), \(n_3=\alpha_1\alpha_4+(\alpha_1-\alpha_{21}-1)(\alpha_2-1)(\alpha_4-1)-\alpha_4+1\), \(n_4=\alpha_1\alpha_2(\alpha_3-1)+\alpha_{21}(\alpha_2-1)+\alpha_2\) (see [J. Komeda, Tsukuba J. Math. 6, 237–270 (1982; Zbl 0546.14011)]).
Using the above generators, the indispensable binomials of these toric ideals (those appearing in every minimal generating set of \(I_S\)) are determined and the monomial algebras having strongly indispensable minimal graded free resolutions are characterized in the first part of the work.
Let \(C_S\) be the affine curve with parametrization \(X_1=t^{n_1}\), \(X_2=t^{n_2}\), \(X_3=t^{n_3}\), \(X_4=t^{n_4}\). The last part of this paper is devoted to give a characterization of when the tangent cone of \(C_S\) is Cohen-Macaulay in terms of the defining integers \(\alpha_i\) and \(\alpha_{21}\).

MSC:

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14H20 Singularities of curves, local rings
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)

Citations:

Zbl 0546.14011

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References:

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