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  "body": "\n## Problem\n\nLet $g \\ge 1$, and let $\\mathcal{A}_g$ be the moduli space of principally polarized complex [[abelian varieties|R7W03TXRS9]] of dimension $g$.\nThe [[Jacobian variety|ZQ84M9D9TC]] of a smooth projective complex curve of genus $g$, with its theta divisor, is a point of $\\mathcal{A}_g$.\nLet $\\mathcal{J}_g \\subseteq \\mathcal{A}_g$ be the set of these points.\nCharacterize the points of $\\mathcal{J}_g$, or of its closure, among all points of $\\mathcal{A}_g$.\n\n## Remarks\n\nBy the [[Torelli theorem|BRKFX8BQ60]], a curve is determined by its Jacobian with its polarization, so $\\mathcal{J}_g$ has the dimension $3g - 3$ of the moduli of curves when $g \\ge 2$.\nThe space $\\mathcal{A}_g$ has dimension $g(g + 1)/2$.\nFor $g \\le 3$ the closure of $\\mathcal{J}_g$ is all of $\\mathcal{A}_g$, and for $g \\ge 4$ it is a proper closed subset.\nFor $g = 4$, Schottky found a modular form, built from [[theta functions|2F5CYQJVAF]], that vanishes on $\\mathcal{J}_4$.\nIgusa showed that its zero locus is exactly the closure of $\\mathcal{J}_4$.\nIn 1986 Shiota proved a conjecture of Novikov on this problem.\nIt says that an indecomposable principally polarized abelian variety is a Jacobian if and only if its theta function solves the KP equation.\nOther characterizations, for example by trisecants of the Kummer variety, are also known.\n\n## Explanations\n\n### L1 human\n\nEvery curve has an abelian variety attached to it, its Jacobian.\nThere are many more abelian varieties than curves once the dimension is at least $4$.\nThe Schottky problem asks how to recognize, from its equations or its theta function, which abelian varieties come from curves.\n",
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