← 研踪 PaperTrailarXiv:2504.16771 · v3
Projective Variety Recovery from Unknown Linear Projections
首次提交 2025-04-23 22:42
版本更新 2026-11-11 02:26
雷达观测 2026-11-11 13:07
摘要
We study how a smooth irreducible variety $X$ of dimension $n$ and degree $d$ in $\mathbb{P}^m$, $m\geq n+2$, can be recovered from the equations of its images under two generic projections from unknown centers. This extends earlier results on curves in $\mathbb{P}^3$ and surfaces in $\mathbb{P}^4$ to any dimension. Since $X$ is the normalization of each image, one image determines $X$ modulo the projective group when $X$ is linearly normal; what two images add is the configuration, that is the pair of projections. For complete intersections we prove unconditionally that the set of configurations compatible with the two images and their Hilbert polynomial is finite, of size at most $|\mathrm{Lin}(X)|$, the projective automorphism group of $X$. The argument is effective and the bound is attained. We then introduce an equation generalizing Kruppa's equation, relating the duals of the images to a fundamental matrix, encoding the pair of projections modulo the projective group. We give a necessary condition for its solution variety to be finite, a bound on its isolated points, and a sufficient condition under which the true fundamental matrix is rigid: no quadric may contain a generic line $L$ together with the locus of points of $X$ whose tangent space meets $L$. Rigidity is unconditional for curves; in higher dimension it holds under a first-order liftability hypothesis, which we certify for a del Pezzo quartic surface in $\mathbb{P}^4$. Global finiteness of that solution variety remains open, necessarily so, as it contains solutions realized by no variety; for $\dim X<\mathrm{codim}\,X-1$ the obstruction amounts to knowing the fundamental matrix itself. Once the projections are known, $X$ is recovered from the two cones over the images: in codimension two their intersection is $X$ plus a component of degree $d(d-1)$; in higher codimension $X$ is its unique component of maximal dimension.
作者备注：arXiv admin note: text overlap with arXiv:math/0208099, arXiv:math/0110157 by other authors
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v4 · 2026-11-11 18:34 · 最新观测版本
Projective Variety Recovery from Unknown Linear Projections
We study how a smooth irreducible variety $X$ of dimension $n$ and degree $d$ in $\mathbb{P}^m$, $m\geq n+2$, can be recovered from the equations of its images under two generic projections from unknown centers. This extends earlier results on curves in $\mathbb{P}^3$ and surfaces in $\mathbb{P}^4$ to any dimension. Since $X$ is the normalization of each image, one image determines $X$ modulo the projective group when $X$ is linearly normal; what two images add is the configuration, that is the pair of projections. For complete intersections we prove unconditionally that the set of configurations compatible with the two images and their Hilbert polynomial is finite, of size at most $|\mathrm{Lin}(X)|$, the projective automorphism group of $X$. The argument is effective and the bound is attained. We then introduce an equation generalizing Kruppa's equation, relating the duals of the images to a fundamental matrix, encoding the pair of projections modulo the projective group. We give a necessary condition for its solution variety to be finite, a bound on its isolated points, and a sufficient condition under which the true fundamental matrix is rigid: no quadric may contain a generic line $L$ together with the locus of points of $X$ whose tangent space meets $L$. Rigidity is unconditional for curves; in higher dimension it holds under a first-order liftability hypothesis, which we certify for a del Pezzo quartic surface in $\mathbb{P}^4$. Global finiteness of that solution variety remains open, necessarily so, as it contains solutions realized by no variety; for $\dim X<\mathrm{codim}\,X-1$ the obstruction amounts to knowing the fundamental matrix itself. Once the projections are known, $X$ is recovered from the two cones over the images: in codimension two their intersection is $X$ plus a component of degree $d(d-1)$; in higher codimension $X$ is its unique component of maximal dimension.
查看此版本解读 →查看此版本原文 ↗v3 · 2026-11-11 02:26
Projective Variety Recovery from Unknown Linear Projections
We study how a smooth irreducible variety $X$ of dimension $n$ and degree $d$ in $\mathbb{P}^m$, $m\geq n+2$, can be recovered from the equations of its images under two generic projections from unknown centers. This extends earlier results on curves in $\mathbb{P}^3$ and surfaces in $\mathbb{P}^4$ to any dimension. Since $X$ is the normalization of each image, one image determines $X$ modulo the projective group when $X$ is linearly normal; what two images add is the configuration, that is the pair of projections. For complete intersections we prove unconditionally that the set of configurations compatible with the two images and their Hilbert polynomial is finite, of size at most $|\mathrm{Lin}(X)|$, the projective automorphism group of $X$. The argument is effective and the bound is attained. We then introduce an equation generalizing Kruppa's equation, relating the duals of the images to a fundamental matrix, encoding the pair of projections modulo the projective group. We give a necessary condition for its solution variety to be finite, a bound on its isolated points, and a sufficient condition under which the true fundamental matrix is rigid: no quadric may contain a generic line $L$ together with the locus of points of $X$ whose tangent space meets $L$. Rigidity is unconditional for curves; in higher dimension it holds under a first-order liftability hypothesis, which we certify for a del Pezzo quartic surface in $\mathbb{P}^4$. Global finiteness of that solution variety remains open, necessarily so, as it contains solutions realized by no variety; for $\dim X<\mathrm{codim}\,X-1$ the obstruction amounts to knowing the fundamental matrix itself. Once the projections are known, $X$ is recovered from the two cones over the images: in codimension two their intersection is $X$ plus a component of degree $d(d-1)$; in higher codimension $X$ is its unique component of maximal dimension.
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