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About 2 weeks ago, I posted a question about irreducibility of a scheme over a completed local ring, on whether this is a continuous property or a limit property, related to a research problem I was working on. While I didn't succeed in fully answering the old question, I somehow got a bit more elementary and fundamental question, for which I was not able to get a rigorous proof. And I guess it should have been known already, as this is such a basic question, but I had difficulties in locating a good reference. So, let me ask.

[The question was corrected a bit reflecting comments.]

To motivate, suppose $k$ is a field of characteristic $0$ (or you may assume something more general), consider the following equation: Let $y_1, y_2, y_3$ be variables, and for nonzero constants $a_1, \cdots, a_4 \in k$, consider the equation $V_{\alpha_0}: a_1 y_1 + a_2 y_1 y_2 + a_3 y_1 y_3^2 + a_4 = 0.$ The shape of the equation does not matter, but it is a finite linear combination of monomials in $y_i$. Roughly put, the question is this. Suppose the affine $k$-scheme $V_{\alpha_0}$ is integral. If we take ``small changes" of $a_i$ to obtain a new affine scheme $V_{\alpha}$, then is $V_{\alpha}$ at least irreducible?

Here, it is important that we do not turn a ``monomial" with $0$ coefficient into something nonzero, i.e. we modify only the coefficients that are nonzero.

One way I was trying to do was to imitate arguments involving incidence varieties to reformulate the question as follows: replace the nonzero constants $a_1, \cdots, a_4$ by variables $x_1, \cdots, x_4$, and consider the general equation $V: x_1 y_1 + x_2 y_1 y_2 + x_3 y_1 y_3 ^2 + x_4 = 0$ in $\mathbb{A}^4 \times \mathbb{A}^3$ (with $(x_1, \cdots, x_4, y_1, y_2, y_3)$ as the coordinates). Consider the projection $pr_1: V \to \mathbb{A}^4$ to the $x$-coordinates, and we are given that for $\alpha_0= (a_1, \cdots, a_4) \in \mathbb{A}^4$, the fiber $V_{\alpha_0} = pr^{-1} (\alpha_0)$ is integral.

Then what I'm asking for is whether one can find an open neighborhood $U \subset \mathbb{A}^4$ of $\alpha_0$ such that for each $\alpha \in U$, the fiber $V_{\alpha} = pr^{-1} (\alpha)$ is irreducible.

I wonder if anyone has a good argument to prove/disprove the above statement. Any suggestions or ideas or discussions would be appreciated.

The situation I'm eventually interested in is the case when I'm given a system of algebraic equations, which can be easily formulated imitating the above single equation case.

About 2 weeks ago, I posted a question about irreducibility of a scheme over a completed local ring, on whether this is a continuous property or a limit property. I didn't succeed in answering it, but I got a bit more elementary question. I guess it should have been known already, as this is a basic question, but I had difficulties in locating a good reference. So, let me ask.

[The question was corrected a bit reflecting comments.]

To motivate, suppose $k$ is a field of characteristic $0$ (or something more general). Let $y_1, y_2, y_3$ be variables, and for nonzero constants $a_1, \cdots, a_4 \in k$, consider the equation $V_{\alpha_0}: a_1 y_1 + a_2 y_1 y_2 + a_3 y_1 y_3^2 + a_4 = 0.$ The shape of the equation does not matter, but it is a finite linear combination of monomials in $y_i$. Roughly put, the question is: Suppose the affine $k$-scheme $V_{\alpha_0}$ is integral. If we take ``small changes" of $a_i$ to obtain a new affine scheme $V_{\alpha}$, then is $V_{\alpha}$ at least irreducible?

Here, it is important that we do not turn a ``monomial" with $0$ coefficient into something nonzero, i.e. we modify only the coefficients that are nonzero.

I tried to reformulate the question as follows: replace the nonzero constants $a_1, \cdots, a_4$ by variables $x_1, \cdots, x_4$, and consider the general equation $V: x_1 y_1 + x_2 y_1 y_2 + x_3 y_1 y_3 ^2 + x_4 = 0$ in $\mathbb{A}^4 \times \mathbb{A}^3$ (with $(x_1, \cdots, x_4, y_1, y_2, y_3)$ as the coordinates). Consider the projection $pr_1: V \to \mathbb{A}^4$ to the $x$-coordinates, and we are given that for $\alpha_0= (a_1, \cdots, a_4) \in \mathbb{A}^4$, the fiber $V_{\alpha_0} = pr^{-1} (\alpha_0)$ is integral.

Then I ask whether one can find an open neighborhood $U \subset \mathbb{A}^4$ of $\alpha_0$ such that for each $\alpha \in U$, the fiber $V_{\alpha} = pr^{-1} (\alpha)$ is irreducible.

Any suggestions or ideas or discussions would be appreciated.

The situation I'm eventually interested in is the case when I'm given a system of algebraic equations, for which a similar question can be formulated.

About 2 weeks ago, I posted a question about irreducibility of a scheme over a completed local ring, on whether this is a continuous property or a limit property, related to a research problem I was working on. While I didn't succeed in fully answering the old question, I somehow got a bit more elementary and fundamental question, for which I was not able to get a rigorous proof. And I guess it should have been known already, as this is such a basic question, but I had difficulties in locating a good reference. So, let me ask.

To motivate, suppose $k$ is a field of characteristic $0$ (or you may assume something more general), consider the following equation: Let $y_1, y_2, y_3$ be variables, and for nonzero constants $a_1, \cdots, a_4 \in k$, consider the equation $V_{\alpha_0}: a_1 y_1 + a_2 y_1 y_2 + a_3 y_1 y_3^2 + a_4 = 0.$ The shape of the equation does not matter, but it is a finite linear combination of monomials in $y_i$. Roughly put, the question is this. Suppose the affine $k$-scheme $V_{\alpha_0}$ is irreducible. If we take ``small changes" of $a_i$ to obtain a new affine scheme $V_{\alpha}$, then is $V_{\alpha}$ irreducible?

Here, it is important that we do not turn a ``monomial" with $0$ coefficient into something nonzero, i.e. we modify only the coefficients that are nonzero.

One way I was trying to do was to imitate arguments involving incidence varieties to reformulate the question as follows: replace the nonzero constants $a_1, \cdots, a_4$ by variables $x_1, \cdots, x_4$, and consider the general equation $V: x_1 y_1 + x_2 y_1 y_2 + x_3 y_1 y_3 ^2 + x_4 = 0$ in $\mathbb{A}^4 \times \mathbb{A}^3$ (with $(x_1, \cdots, x_4, y_1, y_2, y_3)$ as the coordinates). Consider the projection $pr_1: V \to \mathbb{A}^4$ to the $x$-coordinates, and we are given that for $\alpha_0= (a_1, \cdots, a_4) \in \mathbb{A}^4$, the fiber $V_{\alpha_0} = pr^{-1} (\alpha_0)$ is irreducible.

Then what I'm asking for is whether one can find an open neighborhood $U \subset \mathbb{A}^4$ of $\alpha_0$ such that for each $\alpha \in U$, the fiber $V_{\alpha} = pr^{-1} (\alpha)$ is irreducible as well.

I wonder if anyone has a good argument to prove/disprove the above statement. Any suggestions or ideas or discussions would be appreciated.

The situation I'm eventually interested in is the case when I'm given a system of algebraic equations, which can be easily formulated imitating the above single equation case.

About 2 weeks ago, I posted a question about irreducibility of a scheme over a completed local ring, on whether this is a continuous property or a limit property, related to a research problem I was working on. While I didn't succeed in fully answering the old question, I somehow got a bit more elementary and fundamental question, for which I was not able to get a rigorous proof. And I guess it should have been known already, as this is such a basic question, but I had difficulties in locating a good reference. So, let me ask.

[The question was corrected a bit reflecting comments.]

To motivate, suppose $k$ is a field of characteristic $0$ (or you may assume something more general), consider the following equation: Let $y_1, y_2, y_3$ be variables, and for nonzero constants $a_1, \cdots, a_4 \in k$, consider the equation $V_{\alpha_0}: a_1 y_1 + a_2 y_1 y_2 + a_3 y_1 y_3^2 + a_4 = 0.$ The shape of the equation does not matter, but it is a finite linear combination of monomials in $y_i$. Roughly put, the question is this. Suppose the affine $k$-scheme $V_{\alpha_0}$ is integral. If we take ``small changes" of $a_i$ to obtain a new affine scheme $V_{\alpha}$, then is $V_{\alpha}$ at least irreducible?

Here, it is important that we do not turn a ``monomial" with $0$ coefficient into something nonzero, i.e. we modify only the coefficients that are nonzero.

One way I was trying to do was to imitate arguments involving incidence varieties to reformulate the question as follows: replace the nonzero constants $a_1, \cdots, a_4$ by variables $x_1, \cdots, x_4$, and consider the general equation $V: x_1 y_1 + x_2 y_1 y_2 + x_3 y_1 y_3 ^2 + x_4 = 0$ in $\mathbb{A}^4 \times \mathbb{A}^3$ (with $(x_1, \cdots, x_4, y_1, y_2, y_3)$ as the coordinates). Consider the projection $pr_1: V \to \mathbb{A}^4$ to the $x$-coordinates, and we are given that for $\alpha_0= (a_1, \cdots, a_4) \in \mathbb{A}^4$, the fiber $V_{\alpha_0} = pr^{-1} (\alpha_0)$ is integral.

Then what I'm asking for is whether one can find an open neighborhood $U \subset \mathbb{A}^4$ of $\alpha_0$ such that for each $\alpha \in U$, the fiber $V_{\alpha} = pr^{-1} (\alpha)$ is irreducible.

I wonder if anyone has a good argument to prove/disprove the above statement. Any suggestions or ideas or discussions would be appreciated.

The situation I'm eventually interested in is the case when I'm given a system of algebraic equations, which can be easily formulated imitating the above single equation case.

about 2 weeks ago, I posted a question about irreducibility of a scheme over a completed local ring, on whether this is a continuous property or a limit property, related to a research problem I was working on. While I didn't succeed in fully answering the old question, I somehow got a bit more elementary and fundamental question, for which I was not able to get a rigorous proof. And I guess it should have been known already, as this is such a basic question, but I had difficulties in locating a good reference. So, let me ask.

To motivate, suppose $k$ is a field of characteristic $0$ (or you may assume something more general), consider the following equation: Let $y_1, y_2, y_3$ be variables, and for nonzero constants $a_1, \cdots, a_4 \in k$, consider the equation $V_{\alpha_0}: a_1 y_1 + a_2 y_1 y_2 + a_3 y_1 y_3^2 + a_4 = 0.$ The shape of the equation does not matter, but it is a finite linear combination of monomials in $y_i$. Roughly put, the question is this. Suppose $V_{\alpha_0}$ is irreducible. If we take ``small changes" of $a_i$ to obtain a new equation $V_{\alpha}$, then is $V_{\alpha}$ irreducible?

Here, it is important that we do not turn a ``monomial" with $0$ coefficient into something nonzero, i.e. we modify only the coefficients that are nonzero.

One way I was trying to do was to imitate arguments involving incidence varieties to reformulate the question as follows: replace the nonzero constants $a_1, \cdots, a_4$ by variables $x_1, \cdots, x_4$, and consider the general equation $V: x_1 y_1 + x_2 y_1 y_2 + x_3 y_1 y_3 ^2 + x_4 = 0$ in $\mathbb{A}^4 \times \mathbb{A}^3$ (with $(x_1, \cdots, x_4, y_1, y_2, y_3)$ as the coordinates). Consider the projection $pr_1: V \to \mathbb{A}^4$ to the $x$-coordinates, and we are given that for $\alpha_0 \in \mathbb{A}^4$, the fiber $V_{\alpha_0} = pr^{-1} (\alpha_0)$ is irreducible.

Then what I'm asking for is whether one can find an open neighborhood $U \subset \mathbb{A}^4$ of $\alpha_0$ such that for each $\alpha \in U$, the fiber $V_{\alpha} = pr^{-1} (\alpha)$ is irreducible as well.

I wonder if anyone has a good argument to prove/disprove the above statement. Any suggestions or ideas or discussions would be appreciated.

The situation I'm eventually interested in is the case when I'm given a system of algebraic equations, which can be easily formulated imitating the above single equation case.

About 2 weeks ago, I posted a question about irreducibility of a scheme over a completed local ring, on whether this is a continuous property or a limit property, related to a research problem I was working on. While I didn't succeed in fully answering the old question, I somehow got a bit more elementary and fundamental question, for which I was not able to get a rigorous proof. And I guess it should have been known already, as this is such a basic question, but I had difficulties in locating a good reference. So, let me ask.

To motivate, suppose $k$ is a field of characteristic $0$ (or you may assume something more general), consider the following equation: Let $y_1, y_2, y_3$ be variables, and for nonzero constants $a_1, \cdots, a_4 \in k$, consider the equation $V_{\alpha_0}: a_1 y_1 + a_2 y_1 y_2 + a_3 y_1 y_3^2 + a_4 = 0.$ The shape of the equation does not matter, but it is a finite linear combination of monomials in $y_i$. Roughly put, the question is this. Suppose the affine $k$-scheme $V_{\alpha_0}$ is irreducible. If we take ``small changes" of $a_i$ to obtain a new affine scheme $V_{\alpha}$, then is $V_{\alpha}$ irreducible?

Here, it is important that we do not turn a ``monomial" with $0$ coefficient into something nonzero, i.e. we modify only the coefficients that are nonzero.

One way I was trying to do was to imitate arguments involving incidence varieties to reformulate the question as follows: replace the nonzero constants $a_1, \cdots, a_4$ by variables $x_1, \cdots, x_4$, and consider the general equation $V: x_1 y_1 + x_2 y_1 y_2 + x_3 y_1 y_3 ^2 + x_4 = 0$ in $\mathbb{A}^4 \times \mathbb{A}^3$ (with $(x_1, \cdots, x_4, y_1, y_2, y_3)$ as the coordinates). Consider the projection $pr_1: V \to \mathbb{A}^4$ to the $x$-coordinates, and we are given that for $\alpha_0= (a_1, \cdots, a_4) \in \mathbb{A}^4$, the fiber $V_{\alpha_0} = pr^{-1} (\alpha_0)$ is irreducible.

Then what I'm asking for is whether one can find an open neighborhood $U \subset \mathbb{A}^4$ of $\alpha_0$ such that for each $\alpha \in U$, the fiber $V_{\alpha} = pr^{-1} (\alpha)$ is irreducible as well.

I wonder if anyone has a good argument to prove/disprove the above statement. Any suggestions or ideas or discussions would be appreciated.

The situation I'm eventually interested in is the case when I'm given a system of algebraic equations, which can be easily formulated imitating the above single equation case.