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The answer is no, in fact there exist examples of non-reduced projective curves which are non smoothable.

Perhaps the easiest example is the double line, i.e. the scheme $X=2L$, where $L$ is a line on a smooth cubic surface in $\mathbb{P}^3$. In fact, one can show that

1. $H^0(X, T^1_X)=0$, so every deformation of $X$ over the dual numbers is locally trivial;
2. $H^1(X, T_X)=0$, so every locally trivial deformation of $X$ is actually trivial.

It follows that $X$ is a rigid scheme. Since $X$ is projective, rigidity implies that for any flat family $Y \to T$ whose central fibre is isomorphic to $X$, nearby fibres are isomorphic to $X$ too (see [Hartshorne, Deformation Theory, Section 5, in particular Exercise 1.10]).

So $X$ provides a counterexamples to your question.

Notice that Sandor's answer and comments provide a different nilpotent structure on $\mathbb{P}^1$, which is instead smoothable.

I do not know whether it is possible to give a counterexample with $X$ affine.

The answer is no, in fact there exist examples of non-reduced projective curves which are non smoothable.

Perhaps the easiest example is the double line, i.e. the scheme $X=2L$, where $L$ is a line on a smooth cubic surface in $\mathbb{P}^3$. In fact, one can show that

1. $H^0(X, T^1_X)=0$, so every deformation of $X$ over the dual numbers is locally trivial;
2. $H^1(X, T_X)=0$, so every locally trivial deformation of $X$ is actually trivial.

It follows that $X$ is a rigid scheme. Since $X$ is projective, rigidity implies that for any flat family $Y \to T$ whose central fibre is isomorphic to $X$, nearby fibres are isomorphic to $X$ too (see [Hartshorne, Deformation Theory, Section 5, in particular Exercise 5.10]).

So $X$ provides a counterexamples to your question.

Notice that Sandor's answer and comments provide a different nilpotent structure on $\mathbb{P}^1$, which is instead smoothable.

I do not know whether it is possible to give a counterexample with $X$ affine.

The answer is no, in fact there exist examples of non-reduced projective curves which are non smoothable.

Perhaps the easiest example is the double line, i.e. the scheme $X=2L$, where $L$ is a line on a smooth cubic surface in $\mathbb{P}^3$. In fact, one can show that

1. $H^0(X, T^1_X)=0$, so every deformation of $X$ over the dual numbers is locally trivial;
2. $H^1(X, T_X)=0$, so every locally trivial deformation of $X$ is actually trivial.

It follows that $X$ is a rigid scheme. Since $X$ is projective, rigidity implies that for any flat family $Y \to T$ whose central fibre is isomorphic to $X$, nearby fibres are isomorphic to $X$ too (see [Hartshorne, Deformation Theory, Section 5, in particular Exercise 1.10]).

So $X$ provides a counterexamples to your question.

Notice that Sandor's answer and comments provide a different nilpotent structure on $L$, which is instead smoothable.

I do not know whether it is possible to give a counterexample with $X$ affine.

The answer is no, in fact there exist examples of non-reduced projective curves which are non smoothable.

Perhaps the easiest example is the double line, i.e. the scheme $X=2L$, where $L$ is a line on a smooth cubic surface in $\mathbb{P}^3$. In fact, one can show that

1. $H^0(X, T^1_X)=0$, so every deformation of $X$ over the dual numbers is locally trivial;
2. $H^1(X, T_X)=0$, so every locally trivial deformation of $X$ is actually trivial.

It follows that $X$ is a rigid scheme. Since $X$ is projective, rigidity implies that for any flat family $Y \to T$ whose central fibre is isomorphic to $X$, nearby fibres are isomorphic to $X$ too (see [Hartshorne, Deformation Theory, Section 5, in particular Exercise 1.10]).

So $X$ provides a counterexamples to your question.

Notice that Sandor's answer and comments provide a different nilpotent structure on $\mathbb{P}^1$, which is instead smoothable.

I do not know whether it is possible to give a counterexample with $X$ affine.

The answer is no, in fact there exist examples of non-reduced projective curves which are non smoothable.

Perhaps the easiest example is the double line, i.e. the scheme $X=2L$, where $L \cong \mathbb{P}^1$. In fact, one can show that

1. $H^0(X, T^1_X)=0$, so every deformation of $X$ over the dual numbers is locally trivial;
2. $H^1(X, T_X)=0$, so every locally trivial deformation of $X$ is actually trivial.

It follows that $X$ is a rigid scheme. Since $X$ is projective, rigidity implies that for any flat family $Y \to T$ whose central fibre is isomorphic to $X$, nearby fibres are isomorphic to $X$ too (see [Hartshorne, Deformation Theory, Section 5]).

So $X$ provides a counterexamples to your question.

I do not know whether it is possible to give a counterexample with $X$ affine.

The answer is no, in fact there exist examples of non-reduced projective curves which are non smoothable.

Perhaps the easiest example is the double line, i.e. the scheme $X=2L$, where $L$ is a line on a smooth cubic surface in $\mathbb{P}^3$. In fact, one can show that

1. $H^0(X, T^1_X)=0$, so every deformation of $X$ over the dual numbers is locally trivial;
2. $H^1(X, T_X)=0$, so every locally trivial deformation of $X$ is actually trivial.

It follows that $X$ is a rigid scheme. Since $X$ is projective, rigidity implies that for any flat family $Y \to T$ whose central fibre is isomorphic to $X$, nearby fibres are isomorphic to $X$ too (see [Hartshorne, Deformation Theory, Section 5, in particular Exercise 1.10]).

So $X$ provides a counterexamples to your question.

Notice that Sandor's answer and comments provide a different nilpotent structure on $L$, which is instead smoothable.

I do not know whether it is possible to give a counterexample with $X$ affine.