# Is it possible to find infinitely many points in a smooth variety such that their dual of corresponding tangent space have nonzero intersection?

Assume $$X\subset \mathbb{P}^n$$ to be a smooth projective variety defined over an algebraic closed field $$k$$ (the characteristic of $$k$$ could be either $$0$$ or positive). For every point $$P\in X$$, denote by $$T_P X$$ its corresponding tangent space. And denote by $$(T_P X)^{\vee}:=\{\text{hyperplanes } H\supset T_P X\}$$ the set of hyperplanes containing $$T_P X$$. In fact, if we consider the dual space $$(\mathbb{P}^n)^{\vee}$$, then $$(T_P X)^{\vee}$$ could be understood as a linear space in it. In particular, $$\dim (T_P X)^{\vee}+\dim X=n-1$$.

My question is, is it possible that there are infinitely many distinct points $$P_1, P_2, \cdots$$ such that $$\bigcap_{i=1}^{\infty} (T_{P_i} X)^{\vee}\neq \emptyset?$$

So far, according to my best knowledge, when $$X$$ is a hypersurface, then the answer to the above question is negative. Since in this case, $$(T_{P_i} X)^{\vee}$$ is exact the Gauss image of $$P_i$$. And by a result of Zak, the Gauss map for a smooth variety is finite. But what will happen if $$X$$ is not a hypersurface? For instance, if $$X$$ is a complete intersection?

Edit: When I first posted the question, I forgot to exclude the obvious examples: the linear subspaces. So maybe I should ask what will happen when the degree of $$X$$ is strictly greater than $$1$$.

• What if $X$ is a straight line?
– user178279
Commented May 19, 2021 at 5:19
• @virkkunen, Thanks. You are right, I should at least exclude this trivial case. I realize that my question is in fact related to the fiber of the conormal map. And your (counter) example is one of the example that the conormal map contract everything. Commented May 19, 2021 at 6:52

You are asking whether there exists a hyperplane in $$\mathbb{P}^n$$ which is tangent to $$X$$ along a positive-dimensional subvariety. Consider the Veronese surface $$V\subset \mathbb{P}^5$$; by definition, the hyperplanes in $$\mathbb{P}^5$$ cut down the full linear system of conics in $$\mathbb{P}^2$$. In particular, there are hyperplanes which intersect $$V$$ along a double line, hence which are tangent to $$V$$ at each point of the line.
• I am not sure I understand the question : as the OP mentions, there are no such examples with hypersurfaces. All you need is a hyperplane section with a singular locus of dimension $>0$ —which can of course be a linear subspace.
• I was thinking of a hyperplane tangent along a line in $X$. Commented May 19, 2021 at 11:53