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Post Closed as "too localized" by S. Carnahan

[ I am a bit clueless about why this question is getting downvotes!? I put it up with a genuine serious interest and I don't seem to be making any egregious error either - apart from those unsure sentences which I have made with a "?" in the bracket. Please explain if something is terribly wrong with this question! Is this question too elementary for this forum? ]

Here by a "curve" I shall tend to think of algebraic curves in $\mathbb{CP}^2$

• Is reducibility or not of a curve a question of whether the defining equation factorizes (necessarily into linear factors?) or is something more demanded from the factors? If the curve is thought of as a monic (in $y$) element in $\mathbb{C}[x][y]$ (which it can always be) has even $1$ root isn't that sufficient to say that it is reducible?

The degree of an algebraic curve will be the highest degree homogeneous component in it and hence if it has a triple point that would imply that the third degree term is the only term. Hence further if this is an ordinary triple point that would mean that this only term (of third degree) has 3 distinct roots and hence the curve is reducible. Is the argument right?

• I would like to understand the other related such statements that I face like - a fourth degree curve with 4 singular (whether or not ordinary? whether or not distinct?) is also reducible, that if a fourth degree irreducible curve has 3 distinct singular points then they are necessarily double.

As the framing itself suggests, I am not sure of the statements and would like to know what is the precise statement that is correct and why.

• Thought of as the monic polynomial (as in the first bullet point) if the curve has $0$ discriminant then it will have repeated roots. Is that then equivalent to saying that irreducibility implies that the discriminant is not identically $0$? (..or is some further work required?..)(I guess the converse is not true - a non-zero discriminant curve can still be reducible?--I guess so..)

• What is the meaning of an "ordinary singular" point on a curve? I am aware of the notion of an "ordinary k-tuple" point. (...is it true that for $k>1$ such a point has to be singular?..seems so..)

• What is the general way to connect reducibility or not of a curve with the fact as to whether or not it has singular points or how many of them does it have?

2 added 3 characters in body

Here by a "curve" I shall tend to think of algebraic curves in $\mathbb{CP}^2$

• Is reducibility or not of a curve a question of whether the defining equation factorizes (necessarily into linear factors?) or is something more demanded from the factors? If the curve is thought of as a monic (in $y$) element in $\mathbb{C}[x][y]$ (which it can always be) has even $1$ root isn't that sufficient to say that it is reducible?

The degree of an algebraic curve will be the highest degree homogeneous component in it and hence if it has a triple point that would imply that the third degree term is the only term. Hence further if this is an ordinary triple point that would mean that this only term (of third degree) has 3 distinct roots and hence the curve is reducible. Is the argument right?

• I would like to understand the other related such statements that I face like - a fourth degree curve with 4 singular (whether or not ordinary? whether or not distinct?) is also reducible, that if a fourth degree irreducible curve has 3 distinct singular points then they are necessarily double.

As the framing itself suggests, I am not sure of the statements and would like to know what is the precise statement that is correct and why.

• Thought of as the monic polynomial (as in the first bullet point) if the curve has $0$ discriminant then it will have repeated roots. Is that then equivalent to saying that irreducibility implies that the discriminant is not identically $0$? (..or is some further work required?..)(I guess the converse is not true - a non-zero discriminant curve can still be reducible?--I guess so..)

• What is the meaning of an "ordinary singular" point on a curve? I am aware of the notion of an "ordinary k-tuple" point. (...is it true that for $k>1$ such a point has to be singular?..seems so..)

• What is the general way to connect reducibility or not of a curve with the fact as to whether or not it has singular points or how many of them does it have?

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