mapped to $(s_1^2+s_2^2: 0:0:-(s_1^2+s_2^2))$. In particular, the base points of $\phi$, $(0:1:i)$ and $(0:1:-i)$, are lying on this line, and the rest of the points are mapped to the point $(1:0:0:-1)$. Outside $L$ at the source and $P$ at the target, $\phi$ is an isomorphism.

Regularity of the symmetric algebra
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Hereafter, we give an upper bound for the regularity of the symmetric algebra ${\mathcal S}_I$ in our setting. We will use in the course of this paper. We begin with some classical notation.

Given a finitely generated graded $S$-module $N$, we will denote by $HF_N$ its Hilbert function and by $HP_N$ its Hilbert polynomial. Recall that for every $\mu$, $$HF_N(\mu ):=\dim_k (N_\mu ),$$ and $$\label{eq:HPHF}
    HP_N (\mu ):=HF_N(\mu )-\sum_i (-1)^i HF_{H_{{\mathfrak m}}^i(N)}(\mu)$$ where ${\mathfrak m}$ denotes the ideal ${\mathfrak m}=(s_0,s_1,s_2)\subset S$. It is well-known that $HF_{H_{{\mathfrak m}}^i(N)}(\mu)$ is finite for any $i$ and $\mu$, and that $HP_N$ is a polynomial function.

We also recall the definition of three classical invariants attached to a graded module $N$. We will denote $$a^i(N):=\sup\{\mu\ |\ H^i_{\mathfrak m}(N)_\mu\neq 0\}, \ \  a^*(N):=\max_i\{a^i(N)\}, \ $$ and $$\operatorname{reg}(N):=\max_i\{a^i(N)+i\}$$ for the $a^i$-invariant, the $a^*$-invariant and the Castelnuovo-Mumford regularity of $N$, respectively. Finally, the notation $\mu(J)$ for a 