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This is a problem that's been bugging me for some time, and therefore I've decided to ask it here. Let $X$ be a smooth projective (irreducible) variety over an algebraically closed field of characteristic 0, let $\mathcal{L},\mathcal{M}\in\mathrm{Pic}(X)$ be two line bundles on $X$, let $Q\subseteq|\mathcal{M}|$ be a linear system of dimension $n\geq1$, let $\mathrm{Div}_{\geq0}(X)$ be the scheme of effective divisors on $X$, let $\mathrm{Aut}^0(X)$ denote the component of the automorphism group scheme of $X$ that contains the identity, and consider the maps

$$\alpha:\mathrm{Div}_{\geq0}(X)\to\mathrm{Pic}(X),\hspace{0.5cm}D\mapsto\mathcal{O}_X(D)$$

$$\beta:\mathrm{Aut}^0(X)\to\mathrm{Pic}(X),\hspace{0.5cm} \theta\mapsto \mathcal{L}\otimes\theta^*\mathcal{M}^{-1}.$$

Now consider the map (where the fiber product incorporates the above maps)

$$\Phi:\mathrm{Div}_{\geq0}(X)\times _{\mathrm{Pic}(X)}\mathrm{Aut}^0(X)\to\mathbb{G}(n,|\mathcal{L}|)$$ $$(D, \theta)\mapsto D+\theta^*Q$$

Question: Assuming the fiber product is not empty, is $\Phi$ an embedding, at least locally? Or even more generally, how can we describe the differential of this map?

I have reason to think that in certain "good" situations this is indeed an embedding (i.e., $X$ a smooth projective curve and $Q$ a specific linear system), but I'm interested in this larger context. Any help would be appreciated.

Edit: In the case that interests me, $Q$ is such that $\Phi$ is injective, so I'm really interested only in the local analysis of this map.

Edit #2: By @Libli's example, $\Phi$ does not have to be injective. However, I am still very much interested in being able to calculate the differential of this map at any given point.

This is a problem that's been bugging me for some time, and therefore I've decided to ask it here. Let $X$ be a smooth projective (irreducible) variety over an algebraically closed field of characteristic 0, let $\mathcal{L},\mathcal{M}\in\mathrm{Pic}(X)$ be two line bundles on $X$, let $Q\subseteq|\mathcal{M}|$ be a linear system of dimension $n\geq1$, let $\mathrm{Div}_{\geq0}(X)$ be the scheme of effective divisors on $X$, let $\mathrm{Aut}^0(X)$ denote the component of the automorphism group scheme of $X$ that contains the identity, and consider the maps

$$\alpha:\mathrm{Div}_{\geq0}(X)\to\mathrm{Pic}(X),\hspace{0.5cm}D\mapsto\mathcal{O}_X(D)$$

$$\beta:\mathrm{Aut}^0(X)\to\mathrm{Pic}(X),\hspace{0.5cm} \theta\mapsto \mathcal{L}\otimes\theta^*\mathcal{M}^{-1}.$$

Now consider the map (where the fiber product incorporates the above maps)

$$\Phi:\mathrm{Div}_{\geq0}(X)\times _{\mathrm{Pic}(X)}\mathrm{Aut}^0(X)\to\mathbb{G}(n,|\mathcal{L}|)$$ $$(D, \theta)\mapsto D+\theta^*Q$$

Question: Assuming the fiber product is not empty, what is the differential of $\Phi$ at a point?

This is a problem that's been bugging me for some time, and therefore I've decided to ask it here. Let $X$ be a smooth projective (irreducible) variety over an algebraically closed field of characteristic 0, let $\mathcal{L},\mathcal{M}\in\mathrm{Pic}(X)$ be two line bundles on $X$, let $Q\subseteq|\mathcal{M}|$ be a linear system of dimension $n\geq1$, let $\mathrm{Div}_{\geq0}(X)$ be the scheme of effective divisors on $X$, let $\mathrm{Aut}^0(X)$ denote the component of the automorphism group scheme of $X$ that contains the identity, and consider the maps

$$\alpha:\mathrm{Div}_{\geq0}(X)\to\mathrm{Pic}(X),\hspace{0.5cm}D\mapsto\mathcal{O}_X(D)$$

$$\beta:\mathrm{Aut}^0(X)\to\mathrm{Pic}(X),\hspace{0.5cm} \theta\mapsto \mathcal{L}\otimes\theta^*\mathcal{M}^{-1}.$$

Now consider the map (where the fiber product incorporates the above maps)

$$\Phi:\mathrm{Div}_{\geq0}(X)\times _{\mathrm{Pic}(X)}\mathrm{Aut}^0(X)\to\mathbb{G}(n,|\mathcal{L}|)$$ $$(D, \theta)\mapsto D+\theta^*Q$$

Question: Assuming the fiber product is not empty, is $\Phi$ an embedding, at least locally? Or even more generally, how can we describe the differential of this map?

I have reason to think that in certain "good" situations this is indeed an embedding (i.e., $X$ a smooth projective curve and $Q$ a specific linear system), but I'm interested in this larger context. Any help would be appreciated.

Edit: In the case that interests me, $Q$ is such that $\Phi$ is injective, so I'm really interested only in the local analysis of this map.

This is a problem that's been bugging me for some time, and therefore I've decided to ask it here. Let $X$ be a smooth projective (irreducible) variety over an algebraically closed field of characteristic 0, let $\mathcal{L},\mathcal{M}\in\mathrm{Pic}(X)$ be two line bundles on $X$, let $Q\subseteq|\mathcal{M}|$ be a linear system of dimension $n\geq1$, let $\mathrm{Div}_{\geq0}(X)$ be the scheme of effective divisors on $X$, let $\mathrm{Aut}^0(X)$ denote the component of the automorphism group scheme of $X$ that contains the identity, and consider the maps

$$\alpha:\mathrm{Div}_{\geq0}(X)\to\mathrm{Pic}(X),\hspace{0.5cm}D\mapsto\mathcal{O}_X(D)$$

$$\beta:\mathrm{Aut}^0(X)\to\mathrm{Pic}(X),\hspace{0.5cm} \theta\mapsto \mathcal{L}\otimes\theta^*\mathcal{M}^{-1}.$$

Now consider the map (where the fiber product incorporates the above maps)

$$\Phi:\mathrm{Div}_{\geq0}(X)\times _{\mathrm{Pic}(X)}\mathrm{Aut}^0(X)\to\mathbb{G}(n,|\mathcal{L}|)$$ $$(D, \theta)\mapsto D+\theta^*Q$$

Question: Assuming the fiber product is not empty, is $\Phi$ an embedding, at least locally? Or even more generally, how can we describe the differential of this map?

I have reason to think that in certain "good" situations this is indeed an embedding (i.e., $X$ a smooth projective curve and $Q$ a specific linear system), but I'm interested in this larger context. Any help would be appreciated.

Edit: In the case that interests me, $Q$ is such that $\Phi$ is injective, so I'm really interested only in the local analysis of this map.

Edit #2: By @Libli's example, $\Phi$ does not have to be injective. However, I am still very much interested in being able to calculate the differential of this map at any given point.

This is a problem that's been bugging me for some time, and therefore I've decided to ask it here. Let $X$ be a smooth projective (irreducible) variety over an algebraically closed field of characteristic 0, let $\mathcal{L},\mathcal{M}\in\mathrm{Pic}(X)$ be two line bundles on $X$, let $Q\subseteq|\mathcal{M}|$ be a linear system of dimension $n\geq1$, let $\mathrm{Div}_{\geq0}(X)$ be the scheme of effective divisors on $X$, let $\mathrm{Aut}^0(X)$ denote the component of the automorphism group scheme of $X$ that contains the identity, and consider the maps

$$\alpha:\mathrm{Div}_{\geq0}(X)\to\mathrm{Pic}(X),\hspace{0.5cm}D\mapsto\mathcal{O}_X(D)$$

$$\beta:\mathrm{Aut}^0(X)\to\mathrm{Pic}(X),\hspace{0.5cm} \theta\mapsto \mathcal{L}\otimes\theta^*\mathcal{M}^{-1}.$$

Now consider the map (where the fiber product incorporates the above maps)

$$\Phi:\mathrm{Div}_{\geq0}(X)\times _{\mathrm{Pic}(X)}\mathrm{Aut}^0(X)\to\mathbb{G}(n,|\mathcal{L}|)$$ $$(D, \theta)\mapsto D+\theta^*Q$$

Question: Assuming the fiber product is not empty, is $\Phi$ an embedding, at least locally? Or even more generally, how can we describe the differential of this map?

I have reason to think that in certain "good" situations this is indeed an embedding (i.e., $X$ a smooth projective curve), but I'm interested in this larger context. Any help would be appreciated.

This is a problem that's been bugging me for some time, and therefore I've decided to ask it here. Let $X$ be a smooth projective (irreducible) variety over an algebraically closed field of characteristic 0, let $\mathcal{L},\mathcal{M}\in\mathrm{Pic}(X)$ be two line bundles on $X$, let $Q\subseteq|\mathcal{M}|$ be a linear system of dimension $n\geq1$, let $\mathrm{Div}_{\geq0}(X)$ be the scheme of effective divisors on $X$, let $\mathrm{Aut}^0(X)$ denote the component of the automorphism group scheme of $X$ that contains the identity, and consider the maps

$$\alpha:\mathrm{Div}_{\geq0}(X)\to\mathrm{Pic}(X),\hspace{0.5cm}D\mapsto\mathcal{O}_X(D)$$

$$\beta:\mathrm{Aut}^0(X)\to\mathrm{Pic}(X),\hspace{0.5cm} \theta\mapsto \mathcal{L}\otimes\theta^*\mathcal{M}^{-1}.$$

Now consider the map (where the fiber product incorporates the above maps)

$$\Phi:\mathrm{Div}_{\geq0}(X)\times _{\mathrm{Pic}(X)}\mathrm{Aut}^0(X)\to\mathbb{G}(n,|\mathcal{L}|)$$ $$(D, \theta)\mapsto D+\theta^*Q$$

Question: Assuming the fiber product is not empty, is $\Phi$ an embedding, at least locally? Or even more generally, how can we describe the differential of this map?

I have reason to think that in certain "good" situations this is indeed an embedding (i.e., $X$ a smooth projective curve and $Q$ a specific linear system), but I'm interested in this larger context. Any help would be appreciated.

Edit: In the case that interests me, $Q$ is such that $\Phi$ is injective, so I'm really interested only in the local analysis of this map.