Boundary-component conjecture for the extended gradient zero locus

Let AgPerf(4,8){\mathcal A}_g^{\operatorname{Perf}}(4,8) be the perfect-cone compactification with level (4,8)(4,8) structure, let fm\overline{f_m} be the extension of the section fmf_m to this compactification, and let {fm=0}\{\overline{f_m}=0\} denote its zero locus. The boundary is the complement of the open moduli space Ag(4,8){\mathcal A}_g(4,8).

Boundary-component conjecture. The locus

{fm=0}AgPerf(4,8)\{\overline{f_m}=0\}\subset{{\mathcal A}_g^{\operatorname{Perf}}}(4,8)

has no irreducible components contained in the boundary.

This conjecture would ensure that the zero locus of the extended sections has no extraneous components created entirely at the boundary, so that it reflects the closure of I(g)I^{(g)}. The source gives no resolution status.

Sources & referencesView supporting material

Primary source

Samuel Grushevsky and Klaus Hulek, “Geometry of theta divisors — a survey”, arXiv:1204.2734 (2013).

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