Let $Z\subset \mathbb{P}^n$ be a pure 1-dim closed subscheme and $n\geq 2$. We assume that $\deg Z:=\mathbb{P}^{n-1}.Z=d$ and $\dim T_p Z=d$ for any closed point $p\in Z$.
Is there any classification of such $Z$, and any restriction of the possible value of $\chi(\mathcal{O}_Z)$?
When $d=1$, $Z$ is a line. How about $d>1$?
If $Z$ is a pure one-dimensional closed subscheme of degree $d$ in projective space, with $d$-dimensional tangent space at every closed point, then $Z$ is supported on a line.
To see this, let $H$ be a general hyperplane. At each point of $H \cap Z$, the tangent space to $H \cap Z$ at that point is the intersection of the hyperplane $H$ with the $d$-dimensional tangent space to $Z$, so it has dimension at least $d-1$.
A scheme supported at a single point with a $k$-dimensional tangent space has degree at least $k+1$. (Let $A$ be a local ring with maximal ideal $\mathfrak{m}$, over a field $K$. Then, counting dimension as a $K$ vector space, $\deg A = \dim_K A = \dim_K A/\mathfrak{m} + \dim_K \mathfrak{m}/\mathfrak{m}^2 + \dotsb \geq 1+k$.)
So each point of $H \cap Z$ has degree at least $d$. But if $H$ is a general hyperplane then the total degree of $H \cap Z$, adding over all points, is $d$. Therefore $H \cap Z$ consists of (is supported at) just one point. This shows that $Z$ is supported on a line.
The tangent bundle of $Z$ gives a rank $d-1$ subbundle of the normal bundle of the line on which $Z$ is supported (after quotienting out the tangent bundle of the line itself). It seems to me that $Z$ is determined by that rank $d-1$ subbundle, and conversely, any rank $d-1$ subbundle can occur. But on these points I am a little uncertain. I would be grateful if one of the experts here would help with this.
