# Regarding local complete intersection morphism

Following Qing Liu's book Algebraic Geometry and Arithmetic Curves, section 6.3.2, we say that a morphism $$f: X\rightarrow Y$$ of varieties over some field $$k$$ is local complete intersection at $$x\in X$$ if there exists a neighborhood $$U$$ of $$x$$ and a chain of morphisms $$U\overset{i}{\hookrightarrow} Z\overset{g}{\rightarrow} Y$$, where $$g\circ i = f|_U$$, $$Z$$ is some scheme over $$k$$, $$i$$ is a regular embedding and $$g$$ is a smooth morphism. We say $$f$$ is local complete intersection (l.c.i for short) if it is l.c.i at each $$x\in X$$.

Now, let me come to my problem. Let $$X$$ be a smooth complex projective variety, $$Y$$ be another complex projective variety (not necessarily smooth), and let $$f:X\rightarrow Y$$ be a morphism. Let $$Z\subset Y$$ be a smooth closed subvariety. Let $$f : X\rightarrow Y$$ be a morphism with the property that $$f$$ is an isomorphism over $$Y\setminus Z$$, and moreover $$f$$ is a $$\mathbb{P}^n$$- bundle over $$Z$$ for some $$n$$.

I believe that $$f$$ is a l.c.i morphism, for which I have got an argument, but I'm not completely sure if it's correct. Can someone please go through my argument and let me know if it makes sense or not?

Claim: $$f$$ is a l.c.i morphism.

Proof: According to Stacks project {https://stacks.math.columbia.edu/tag/069N}, being l.c.i morphism is fpqc-local on the base, and since any zariski covering is also an fpqc covering, enough to show that $$f$$ is l.c.i over an affine covering of $$Y$$. We consider two cases:

1) If $$y\in Z$$, choose an affine neighborhood $$U$$ of $$y$$ such that $$f^{-1}(U)\cong U\times \mathbb{P}^n$$ and $$f|_{f^{-1}(U)}$$ is the usual projection onto 1st component (recall that $$f$$ is a projective bundle over $$Z$$). Since $$\mathbb{P}^n$$ is l.c.i over $$\mathbb{C}$$, and $$U$$ is flat over $$\mathbb{C}$$, we get that $$U\times \mathbb{P}^n\rightarrow U$$ is also l.c.i by the fact that a flat base change of a l.c.i morphism is a l.c.i morphism (see Stacks Project, https://stacks.math.columbia.edu/tag/069I). Hence we conclude that $$f^{-1}(U)\rightarrow U$$ is l.c.i map.

2) If $$y\in Y\setminus Z =: V$$, then cover $$V$$ by affine opens, and since $$f$$ is isomorphism over $$V$$, clearly $$f$$ is a l.c.i map over each of the open affines.

Hence we conclude that $$f$$ is l.c.i morphism. (proved)

Can someone please let me know if this is correct? Thanks in advance.

• That is not correct. Consider either of the small resolutions of an ordinary threefold double point. This morphism satisfies your hypotheses, but the morphism is not a local complete intersection morphism: the conormal sheaf of the graph of the morphism has rank 3 at every point of the exceptional curve (it would have rank 2 if the morphism were LCI). Jul 15, 2019 at 10:46
• *Typo correction: ... the conormal sheaf of the graph of the morphism has rank $4$ at every point of the exceptional curve (it would have rank $3$ if the morphism were LCI). Jul 15, 2019 at 11:09