Let X be a finite set of points in P^n, and let I_ X denote its homogeneous ideal. Write alpha(I_ X) for the least degree of a nonzero homogeneous element of I_ X and hat alpha(I_ X) for its Waldschmidt constant. Chudnovsky's conjecture. hat alpha(I_ X)≥ fracalpha(I_ X)+n-1n. This conjecture gives a general lower bound for the Waldschmidt constant of a finite set of points in projective space. In the paper, the bound is proved for a standard Bbbk-configuration in P^2; the general statement is not resolved here.

Chudnovsky's conjecture for finite sets of points

Let $\mathbb X$ be a finite set of points in $\mathbb P^n$ , and let $I_{\mathbb X}$ denote its homogeneous ideal. Write $\alpha(I_{\mathbb X})$ for the least degree of a nonzero homogeneous element of $I_{\mathbb X}$ and $\hat\alpha(I_{\mathbb X})$ for its Waldschmidt constant.

Chudnovsky's conjecture.

\hat\alpha(I_{\mathbb X})\geq \frac{\alpha(I_{\mathbb X})+n-1}{n}.

This conjecture gives a general lower bound for the Waldschmidt constant of a finite set of points in projective space. In the paper, the bound is proved for a standard $\Bbbk$ -configuration in $\mathbb P^2$ ; the general statement is not resolved here.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Chudnovsky's conjecture for finite sets of points
Let $X$ be a finite set of points in $\mathbb{P}_{\mathbb{C}}^N$ . Write $\alpha(X)=\alpha_1(X)$ for the minimum degree of a hypersurface containing $X$ , and let $\alpha_m(X)$ be the minimum degree of a hypersurface passing through every point of $X$ with multiplicity at least $m$ . Chudnovsky's conjecture. For every $m\geq 1$ ,
$\frac{\alpha_m(X)}{m}\geq\frac{\alpha(X)+N-1}{N}.$
The conjecture generalizes Chudnovsky's inequality in the projective plane and gives an effective lower bound for interpolation degrees, with consequences for Schwarz exponents. The supplied text does not state its resolution status.
source: Louiza Fouli, Paolo Mantero and Yu Xie, “Chudnovsky's Conjecture for very general points in P_k^N”, arXiv:1604.02217 (2017).

Sources & referencesView supporting material

Primary source

Maria Virginia Catalisano, Giuseppe Favacchio, Elena Guardo and Yong-Su Shin, “The Waldschmidt constant of a standard -configuration in P^2”, arXiv:2307.06014 (2023).

Additional references

12 papers in this index state this conjecture (2011–2023). The statement above is taken from the most recent of them; the others are arXiv:2101.09762, arXiv:2101.12308, arXiv:2009.05022, arXiv:2004.11213, arXiv:1811.02051, arXiv:1705.09946, arXiv:1701.04848, arXiv:1604.02217, arXiv:1310.3552, arXiv:1309.5082, arXiv:1105.0258.

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Chudnovsky's conjecture for finite sets of points

Equivalent formulations 1

Chudnovsky's conjecture for finite sets of points

Primary source

Additional references

Progress summary

Solutions 0