Nagata's conjecture on plane curves through very general points

Theorem 1.1: for every $r\ge10$ there is a countable union $E_r$ of proper Zariski-closed subsets of the configuration space, with nonempty complement, such that for every $(p_1,\dots,p_r)\notin E_r$, every nonzero effective plane curve $C$ of degree $d\ge1$ and all $m_i\le\mathrm{mult}_{p_i}C$, one has $\sum m_i<d\sqrt r$. Reducible and nonreduced curves and unequal multiplicities are included. Section 6 deduces that the multipoint Seshadri constant of a line is $1/\sqrt r$ and extends the theorem to every uncountable algebraically closed field of characteristic zero. It does not treat positive characteristic in this manuscript, and it does not address the SHGH conjecture or other finer statements about linear systems.

Theorem 1.1: for every $r\ge10$ there is a countable union $E_r$ of proper Zariski-closed subsets of the configuration space, with nonempty complement, such that for every $(p_1,\dots,p_r)\notin E_r$, every nonzero effective plane curve $C$ of degree $d\ge1$ and all $m_i\le\mathrm{mult}_{p_i}C$, one has $\sum m_i<d\sqrt r$. Reducible and nonreduced curves and unequal multiplicities are included. Section 6 deduces that the multipoint Seshadri constant of a line is $1/\sqrt r$ and extends the theorem to every uncountable algebraically closed field of characteristic zero. It does not treat positive characteristic in this manuscript, and it does not address the SHGH conjecture or other finer statements about linear systems.