On the smoothness of blowups for certain monomial curves (Q664200)

Let \(m_0<\dots<m_{p+1}\) be a sequence of positive integers which form a minimal arithmetical progression and \(K\) be a field. Let \(C\subset {\mathbf A}_K^{p+2}\) be the monomial curve defined parametrically by \(X_0=T^{m_0},\dots, X_p=T^{m_p},Y=T^{m_{p+1}}\) and \(q\subset R=K[X_0,\dots,X_p,Y]\) be the ideal defining \(C\). Here it shows that the blowup scheme Proj \(R[qt]\) is not smooth for \(p>1\). This shows Francia's conjecture in the above case. Francia's conjecture says that a dimension one prime ideal in a regular local ring is a complete intersection if it has a smooth blowup. In general, this does not hold as \textit{M. R. Johnson} and \textit{S. Morey} shows [J. Pure Appl. Algebra 162, No. 2--3, 303--313 (2001; Zbl 1051.13008)].