Chisini's conjecture on generic coverings of the projective plane

At least 27 years old · documented by

Let B⊂PC2B\subset\mathbb{P}^2_{\mathbb{C}} be a cuspidal branch curve, and let

f:S⟶PC2,f′:S′⟶PC2f:S\longrightarrow\mathbb{P}^2_{\mathbb{C}},\qquad f':S'\longrightarrow\mathbb{P}^2_{\mathbb{C}}

be generic coverings of degree at least 55, meaning that their branch curve has only nodes and cusps and the local monodromy at a smooth point of the branch curve is a transposition. Two such coverings are equivalent when they are equivalent as coverings of PC2\mathbb{P}^2_{\mathbb{C}}.

Chisini's conjecture. Given two generic coverings f:S⟶PC2f:S\longrightarrow\mathbb{P}^2_{\mathbb{C}} and f′:S′⟶PC2f':S'\longrightarrow\mathbb{P}^2_{\mathbb{C}}, both of degree at least 55, with the same branch curve BB, the coverings ff and f′f' are equivalent.

The conjecture asks whether a generic covering of degree at least 55 is determined by its branch curve. The supplied text gives no resolution or qualification of its status.

References

Primary source

Fabrizio Catanese, “Differentiable and deformation type of algebraic surfaces, real and symplectic structures”, arXiv:0705.1522 (2007).

Additional references

4 papers in this index state this conjecture (1998–2007). The statement above is taken from the most recent of them; the others are arXiv:math/9807154, arXiv:math/9807153, arXiv:math/9803144.

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.