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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!

PRT
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  • In this thread there is adiscussion of the "tangent cone": https://math.stackexchange.com/questions/4125495/about-the-definition-of-tangent-space-and-tangent-cone – hm2020 Sep 08 '21 at 12:00
  • As you say: When $p$ is a smooth points it follows the tangent cone $C_p\cong T_p(C)$ is the tangent space of $C$ at $p$, and for different smooth points $p\neq q$ it follows the tangent spaces $T_p(C), T_q(C)$ are the fibers of the tangent bundle $T(C)$ of $C$, and they do not intersect: $T_p(C) \cap T_q(C) = \emptyset$. They are not in general embedded into a common ambient space. You should explain to the audience instances where you have observed that these spaces "intersect". – hm2020 Sep 08 '21 at 12:09
  • Thank you for the remark: I should have specified better that when I wrote $X$ projective hypersurface I mean that I am considering $X$ as the zero locus of a homogeneous equation, that gives a certain projective space as the ambient space. In the second part of the question, when I say that $X$ is a surface in $\mathbb{P}^3$ I mean that I fixed a homogeneous polynomial F(x,y,z,w) and the ambient space is $\mathbb{P}^3(x:y:z:w)$. Thank you for asking (I'll add it in the main question) – PRT Sep 08 '21 at 13:11
  • Concerning the first part, I read that question, but I am not able to figure out how to understand something which is valid for any surface with that property. I mean, if I have an equation which gives a surface in $\mathbb{P}^3$, then I can check what happens doing that calculations explicitly, but I am not sure what to do in the general case – PRT Sep 08 '21 at 13:14
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    Maybe you should include a definition of the tangent cone in your post. I have seen a definition using an embedding: If $t\in C$ is a point choose an open affine subscheme $t\in Spec(A)$ where there is an embedding $Spec(A) \subseteq \mathbb{A}^n_k$ with $t:=(0,..,0)$ is the origin. Let $A:=k[x_1,..,x_n]/I)$ and define the ideal $I^$ as the ideal generated by terms $f_m$ where $f:=f_m+f_{m+1}+ \cdots \in I$. Define the tangent cone as $C_t:=Spec(k[x_1,..,x_n]/I^)$. – hm2020 Sep 08 '21 at 13:31
  • This definition appears to depend on an embedding. In order to speak of "intersections", your spaces must "live in the same ambient space". Which embedding have you chosen? – hm2020 Sep 08 '21 at 13:31
  • If $s,t\in C$ you define two tangent cones $C_s, C_t$ via open affine schemes $t\in U_t, s\in U_s$ and embeddings $U_t \subseteq \mathbb{A}^n_k, U_s \subseteq \mathbb{A}^m_k$. How do you do this and ensure that the tangent cones $C_s, C_t$ live in one common ambient space? – hm2020 Sep 08 '21 at 16:59
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    In the link above I define the "global tangent cone" $\pi:T(C):=Spec(Gr(I)) \rightarrow C:=Spec(A)$. It has the property that for any $k$-rational point $t\in C$ it follows $\pi^{-1}(t) \cong C_t$ is the tangent cone, hence $\pi$ has the tangent cone as fibers. If you can embed $T(C) \subseteq \mathbb{A}^n_k$ into an affine space, it may be this construction is useful. – hm2020 Sep 09 '21 at 13:25
  • Thanks for the comments! Before editing the main question: I want to consider the projective tangent cone (as in https://www.dmi.unict.it/frusso/DMI/Geometria_Algebrica_files/Capitolo1.pdf), so I actually begin with an embedded surface $X \subset \mathbb{P}^3$, so the affine $Spec(A)$ will already come with an embedding in $\mathbb{P}^3$, and the cone $C_t$ will be the closure of the image of the affine tangent cone in $\mathbb{P}^3$. 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! – PRT Sep 10 '21 at 08:58

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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.

hm2020
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