Let $X$ be a projective hypersurface. The tangent cone to $X$ at $P$ can be obtained as follows: take an affine open $U$ where $P$ is the origin. Here $X = V(f)$ where $f=f_m+f_{m+1}+...$ is the decomposition of $f$ in homogeneous polynomials. The tangent cone $C_P$ to $X$ at $P$ is the closure of $V(f_m)$. When $P$ is a smooth point of $X$, $C_P$ coincides with the tangent plane to $X$ at $P$. Things becomes more complicated when $P$ is a singular point of $X$.
I wish to understand better the case of X a surface in $\mathbb{P}^3$, namely I fix a homogeneous polynomial F(x,y,z,w) and the ambient space is $\mathbb{P}^3(x:y:z:w)$, and the singular locus of $X$ is a curve. I want to consider the family of tangent cones $\{C_t\}$ , where $t \in X^{sing}$ is general.
Now this is a first broad question:
Can we say something about the common intersection of the elements in $\{C_t\}$? That is, is there a fixed locus $W \subset \mathbb{P}^3$ such that $W \subset C_t$ for $t$ general in $X^{sing}$?
It seems a pretty wild question, at least to me. My impression, for instance, is that $W$ will usually be empty, namely there no point in $\mathbb{P}^3$ belongs to $C_t$ for $t$ general. However I also expect to find example in which the general $C_t$ contains points, or a curve.
For sure there are trivial explicit examples of such a behaviour. For instance: if $X$ is a cone over a curve with a singular point, all the $C_t$ except the one corresponding to the vertex of the cone are the same, so the locus $W$ has dimension $2$.
Let me state a more specific question:
Is it possible to characterize all the surfaces such that there is a fixed $W \subset \mathbb{P}^3$ such that $W \subset C_t$ for $t$ general in $X^{sing}$, and $W$ has dimension 0, 1, 2?
Thank you for your insights!
Question: "Thanks for the comments! Before editing the main question: I want to consider the projective tangent cone (as in dmi.unict.it/frusso/DMI/Geometria_Algebrica_files/Capitolo1.pdf), so I actually begin with an embedded surface $X⊂P^3$, so the affine $Spec(A)$ will already come with an embedding in $P^3$, and the cone Ct will be the closure of the image of the affine tangent cone in $P^3$. Does this clarify what I mean? You are right in the last comment about the global tangent cone. I'll check the link from this perspective, thanks"
Q1:"Can we say something about the common intersection of the elements in ${C_t}$? That is, is there a fixed locus $W⊂P^3$ such that $W⊂C_t$ for $t$ general in $X_{sing}$?"
Anwer: For the question to make any sense you should give a definition of the "global tangent cone" where the fibers $X_t,X_s$ live in the "same ambient variety/scheme": If you want to intersect $X_t \cap X_s$, you must have an embedding $X_r, X_s \subseteq Y$ into some scheme $Y$.
I have not read the paper in your link, but I believe the definition given in the link to my post above (and below) generalize: If $X \subseteq \mathbb{P}^n$ is any quasi projective scheme, there is the "ideal of the diagonal $I_X \subseteq \mathcal{O}_{X \times X}$, and you may construct the relative affine spectrum
$$\pi: T(X):=Spec(Gr(I_X)) \rightarrow X.$$
If $U:=Spec(A)\subseteq X$ is an affine open subset, you should get $\pi^{-1}(U) \cong T(U):=Spec(Gr(I_A))$ is the global tangent cone of $A$. Here $I_A \subseteq A\otimes_k A$ is the ideal of the diagonal of $A$. Hence it seems to me $T(X)$ is well defined and has the tangent cone $X_t$ as fiber for any $k$-rational point $t\in X$. There are details to be verified. If this is well defined you can use the global tangent cone $T(X)$ to study your problem.
Example: If $s,t\in X$ are $k$-rational points, it follows $X_t,X_s \subseteq T(X)$ are closed subschemes and you may intersect $X_s\cap X_t \subseteq T(X)$ inside $T(X)$. Hence the global tangent cone $T(X)$ is the correct object to study for this problem. If $X_{sing} \subseteq X$ is the singular subscheme, you may take the inverse image $\pi^{-1}(X_{sing}) \subseteq T(X)$ and you get a canonical map of schemes
$$\pi_{sing}:\pi^{-1}(X_{sing}) \rightarrow X_{sing},$$
and you may define $W:= \cap_{t\in X_{sing}} X_t \subseteq \pi^{-1}(X_{sing})$.
Q2: "It seems a pretty wild question, at least to me. My impression, for instance, is that W will usually be empty, namely there no point in P3 belongs to Ct for t general. However I also expect to find example in which the general Ct contains points, or a curve."
Answer: Now you have a definition of a "global tangent cone" $T(X)$ and I have constructed an intersection $W \subseteq T(X)$ and you should try to construct non-trivial examples. How does $W$ look like?
Example: If $A:=k[x_1,..,x_n]$ and $X:=Spec(A) \cong \mathbb{A}^n_k$ it follows $T(X) \cong \mathbb{A}^{2n}_k$ is the global tangent cone of $X$. In general if $A$ is a $k$-algebra where $Sym_A^n(\Omega^1_{A/k}) \cong I^n/I^{n+1}$ and $\Omega^1_{A/k}$ is a projective $A$-module, it follows $T(X) \cong \mathbb{V}(\Omega^1_{A/k})\cong T_X$ is the affine vector bundle of $\Omega^1_{A/k}$ - this is (in this case when $\Omega^1_{A/k}$ is projective) the geometric tangent bundle of $X$. This is an algebraic vector bundle on $X$ in the sense of Hartshorne, Chapter II.5. In this case the intersection $W:=\cap_{t\in Z \subseteq X^{cl}} X_t=\emptyset $ will be empty. Different fibers of a vector bundle do not intersect. In fact: If $p\in X_t \cap X_s$ is a $k$-rational point, it follows
$$s=\pi(p)=t$$
hence $s=t$.
Example: If $k$ is the complex number field and $X \subseteq \mathbb{P}^n_k$ is a quasi projective scheme and $\pi: T(X) \rightarrow X$ is the tangent cone in the above sense, it follows $X_t \cap X_s=\emptyset $ if $s \neq t$ are closed points. Hence it follows your scheme $W = \emptyset$. Whenever you have a morphism $f: X\rightarrow Y$ it follows different fibers have empty intersection.
Here is the link to the previous post on the issue:
About the definition of tangent space and tangent cone.