Call a scheme $Y$ proper positive-dimensional over $\mathrm{Spec}\,\mathbb{C}$ **abnormal** if there exists an irreducible scheme $X$ affine of finite type over $\mathrm{Spec}\,\mathbb{C}$ and a $\mathbb{C}$-morphism $X\rightarrow Y$ bijective on the underlying spaces.

Call a scheme $Y$ proper positive-dimensional over $\mathrm{Spec}\,\mathbb{C}$ **strongly abnormal** if there exists an irreducible scheme $X$ affine of finite type over $\mathrm{Spec}\,\mathbb{C}$ and a $\mathbb{C}$-morphism $X\rightarrow Y$ inducing a homeomorphism on the underlying spaces.

Strongly abnormal schemes are obviously abnormal, and abnormal schemes can not be normal (by Zariski's main). By point-set topology, abnormal schemes are irreducible. An example of a strongly abnormal scheme is the projective nodal cubic.

Are there examples of strongly abnormal schemes of each dimension that are not direct products of positive-dimensional $\mathbb{C}$-schemes? Are there examples of abnormal schemes that are not strongly abnormal of each dimension that are not direct products of positive-dimensional $\mathbb{C}$-schemes?

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