Any connected reduced proper $k$-scheme is irreducible? Here $k$ is a field.
2026-04-11 10:30:05.1775903405
Connected reduced proper $k$-scheme
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No, you can take the union of two projective lines that intersect transversely at a point. Concretely, this can be given as $\text{Proj} (k[X,Y,Z]/(XY))$, the union of the $X$- and $Y$-axes in the projective plane $\mathbb{P}^2$.