# In what sense is a generically submersive morphism of varieties subermersive over singular points?

## Background/Motivation

I'm currently interested in the duality theorem for projective varieties and more specifically in properties of the conormal variety over the dual variety.

Let $V$ be a $k$-vector space over a field of characteristic 0. Denote by $V^\ast$ its dual space. Note that a point in the dual projective space $\mathbf{P}(V^\ast)$ is a hyperplane in $\mathbf{P}(V)$.

Given a projective variety $X\subset\mathbf{P}(V)$, the conormal variety is the Zariski closure in $\mathbf{P}(V)\times\mathbf{P}(V^\ast)$ of the set of tuples $(x,[H])$ where $x$ is a regular point of $X$ and $H$ is a hyperplane in $\mathbf{P}(V)$ containing the embedded tangent space to $X$ at $x$. The projection to the factor $\mathbf{P}(V^\ast)$ is (by definition) the dual variety $X^\ast$ to $X$. The duality theorem states, that the dual of $X^\ast$ is again $X$. (see e.g. "Algebraic Geometry - A First Course" by J. Harris, Lecture 15)

An important point of the argument in the proof of the duality theorem in the book "Algebraic Geometry - A First Course" by J. Harris (Lecture 16, Example 16.20) is that the differentials of the projections $\pi_1\colon {\rm CN}(X)\to X$ and $\pi_2\colon {\rm CN}(X)\to X^\ast$ are surjective (for points in the fibre of smooth points $x\in X$ and $[H]\in X^\ast$). I am interested in the singular points of the dual variety.

I have two questions:

Let $X$ and $Y$ be varieties. Assume that $X$ is smooth. Let $f\colon X\to Y$ be a finite morphism and suppose that $f$ is submersive at all points $x\in X$ such that $f(x)$ is a regular point of $Y$. Take a subvariety $W\subset Y$ and $x\in X$ such that $y:=f(x)$ is a regular point of $W$ (and a singular point of $Y$). Is it true that ${\rm d}f_x(T_x(X))\supset T_y(W)$ as subsets of the tangent cone $TC_y(Y)$. If so, what is a reference for that?

Let $X\subset \mathbf{P}^N$, $Y\subset\mathbf{P}^n$ be projective varieties and let $\pi\colon \mathbf{P}^N\to\mathbf{P}^n$ be a linear projection. Assume that $\pi(X)=Y$ and that for all regular points $x\in X$ such that $\pi(x)$ is a regular point of $Y$, we have $\pi(T_x(X))=T_{\pi(x)}(Y)$. Let $W\subset Y$ be a subvariety and $y\in W$ be regular point of $W$. Let $x$ be a regular point of $X$ with $\pi(x)=y$. Is it then true, that $\pi(T_x(X))$ contains the tangent space $T_y(W)$ (as subsets of the tangent cone $TC_y(Y)$.

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Negative answer to first question: let $C \subset \mathbb{A}^2$ be the plane curve (in characteristic $\neq 2,\,3$) with equation $y^2=x^3$, with singular point $Q=(0,0)$, and let $p:\mathbb{A}^1\to C$ be the normalization morphism $t\mapsto (t^2,t^3)$. Now take $X=\mathbb{A}^2$, $Y=C\times \mathbb{A}^1$, and $f: X\to Y$ given by $f(t,u)=(p(t),tu)$. Finally, take $W=\{Q\}\times \mathbb{A}^1$, $x=(0,0)$, $y=(Q,0)$. The differential of $f$ has rank $2$ whenever $t\neq0$ but is zero at $x$.