Existence of a smooth quadratic non-complete-intersection 11-fold in $\mathbb{P}^{17}$ of Fano index 8: the remaining case of Watanabe's classification of quadratic varieties with $n=2c-1$

For a complex nondegenerate smooth projective variety $X \subset \mathbb{P}^{n+c}$ of dimension $n$ that is scheme-theoretically defined by quadratic equations (a quadratic variety), Ionescu and Russo proved Hartshorne's complete-intersection conjecture for $n \geq 2c+1$ and classified the boundary case $n=2c$. Watanabe (SIGMA 2025) classified the next boundary case $n=2c-1$ up to a single undecided possibility: an 11-dimensional Fano variety $X \subset \mathbb{P}^{17}$ of Fano index 8, covered by lines, whose Hilbert scheme of lines through a general point is a Gushel–Mukai 6-fold. The paper derives strong necessary conditions for such a variety (degree $H^{11}=24$, $c_1=8H$, $c_2=31H^2$, conic-connectedness, no double cover onto a smooth quadratic variety in $\mathbb{P}^{16}$) but neither constructs it nor rules it out. The problem is to determine whether this fourth case actually occurs: construct such a variety, or prove it is vacuous.

Background

A smooth projective variety $X \subset \mathbb{P}^{N}$ of dimension $n$ and codimension $c = N-n$ is called a complete intersection if its homogeneous ideal is generated by exactly $c$ polynomials; it is called quadratic if it is scheme-theoretically cut out by quadratic equations (not necessarily $c$ of them). Hartshorne's 1974 conjecture predicts that any smooth nondegenerate (i.e. not contained in any hyperplane) $X \subset \mathbb{P}^{n+c}$ with $n \geq 2c+1$ is a complete intersection; it remains open in general. Ionescu and Russo (2013) proved the conjecture for quadratic varieties and, in the extremal ratio $n = 2c$, showed that the only quadratic varieties that are not complete intersections are the 6-dimensional Grassmannian $G(2,\mathbb{C}^5) \subset \mathbb{P}^{9}$ (parametrizing 2-dimensional linear subspaces of $\mathbb{C}^5$ via the Plücker embedding) and the 10-dimensional spinor variety $\mathbb{S}_{10} \subset \mathbb{P}^{15}$ (an irreducible component of the orthogonal Grassmannian of isotropic 5-planes in $\mathbb{C}^{10}$).

Their method passes to families of lines: a smooth quadratic variety of small codimension is a Fano variety (its anticanonical divisor $-K_X$ is ample), covered by lines (meaning that through every point of $X$ there passes a line entirely contained in $X$), and for a general point $x \in X$ the Hilbert scheme $\mathcal{L}_x$ of lines on $X$ through $x$ (the parameter space of lines on $X$ passing through $x$) embeds via the tangent map into $\mathbb{P}((T_xX)^\vee) \cong \mathbb{P}^{n-1}$ again as a quadratic variety; if $\mathcal{L}_x$ is a complete intersection, so is $X$. The Fano index $i_X$ is the maximal integer $r$ with $-K_X \sim rH$ for an ample generator $H$ of $\mathrm{Pic}(X)$.

Watanabe (2025) extended the classification to the next boundary ratio $n = 2c-1$: a smooth nondegenerate quadratic $X \subset \mathbb{P}^{n+c}$ with $n = 2c-1$ that is not a complete intersection is projectively equivalent to one of four possibilities: (i) the Segre 3-fold $\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$ (the image of $\mathbb{P}^1 \times \mathbb{P}^2$ under the Segre embedding, i.e. the variety of rank-one $2 \times 3$ matrices); (ii) a hyperplane section of $G(2,\mathbb{C}^5) \subset \mathbb{P}^9$; (iii) a hyperplane section of $\mathbb{S}_{10} \subset \mathbb{P}^{15}$; or (iv) an 11-dimensional Fano variety $X \subset \mathbb{P}^{17}$ (so codimension $c = 6$) of Fano index 8, covered by lines, such that $\mathcal{L}_x \subset \mathbb{P}((T_xX)^\vee) \cong \mathbb{P}^{10}$ is a Gushel–Mukai 6-fold, i.e. a smooth quadric section of the cone over $G(2,\mathbb{C}^5) \subset \mathbb{P}^9$ inside $\mathbb{P}^{10}$. Cases (i)–(iii) are realized varieties; case (iv) is only a potential one, and Watanabe states explicitly that he "needs to determine whether the fourth case that appeared in Theorem 1.2 occurs."

Section 5 of that paper derives necessary conditions any case-(iv) variety must satisfy: by the Barth–Larsen theorem (which computes the low-degree integral cohomology of low-codimension subvarieties of projective space) one has $H^2(X,\mathbb{Z}) \cong \mathbb{Z}[H]$ and $H^4(X,\mathbb{Z}) \cong \mathbb{Z}[H^2]$; the Chern classes are $c_1(X) = 8H$ and $c_2(X) = 31H^2$; the degree is $H^{11} = 24$; $X$ is conic-connected (two general points are joined by a conic in $X$); the locus swept by lines through $x$ is isomorphic to the cone over $\mathcal{L}_x$; and $X$ admits no double cover onto a smooth quadratic variety $Y \subset \mathbb{P}^{16}$ (such a cover would force $\deg X = 64$, contradicting $H^{11}=24$). Conversely, Remark 5.5 observes that proving every such $X$ must admit such a double cover would eliminate case (iv) entirely. A related classification of Ionescu and Russo (2010) handles conic-connected manifolds fully only when the Picard number is 2, so it does not settle this Picard-number-1 case.

Problem Statement

Does there exist a complex nondegenerate smooth projective variety $X \subset \mathbb{P}^{17}$ of dimension 11 that is scheme-theoretically defined by quadratic equations and is not a complete intersection? By Watanabe's Theorem 1.2 this is equivalent to asking whether case (iv) of the classification occurs: a smooth 11-dimensional Fano variety $X \subset \mathbb{P}^{17}$ of Fano index 8, covered by lines, such that for a general point $x \in X$ the Hilbert scheme $\mathcal{L}_x$ of lines on $X$ through $x$, embedded in $\mathbb{P}((T_xX)^\vee) \cong \mathbb{P}^{10}$ by the tangent map, is a Gushel–Mukai 6-fold. Any such $X$ must satisfy $c_1(X) = 8H$, $c_2(X) = 31H^2$, $H^{11} = 24$, be conic-connected, and admit no double cover onto a smooth quadratic variety in $\mathbb{P}^{16}$. Decide the question in either direction: construct such a variety (with proof of its listed properties), or prove that no such variety exists, thereby completing the classification of quadratic varieties with $n = 2c-1$.

References

1. Kiwamu Watanabe, "Quadratic Varieties of Small Codimension", Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 21 (2025), 045, 14 pages, DOI 10.3842/SIGMA.2025.045, arXiv:2405.04002, https://arxiv.org/abs/2405.04002
2. Paltin Ionescu and Francesco Russo, "Manifolds covered by lines and the Hartshorne Conjecture for quadratic manifolds", American Journal of Mathematics 135 (2013), no. 2, 349–360, DOI 10.1353/ajm.2013.0020, arXiv:1209.2047, https://arxiv.org/abs/1209.2047
3. Paltin Ionescu and Francesco Russo, "Conic-connected manifolds", Journal für die reine und angewandte Mathematik 644 (2010), 145–157, DOI 10.1515/crelle.2010.054, arXiv:math/0701885, https://arxiv.org/abs/math/0701885
4. Robin Hartshorne, "Varieties of small codimension in projective space", Bulletin of the American Mathematical Society 80 (1974), 1017–1032, DOI 10.1090/S0002-9904-1974-13612-8, https://doi.org/10.1090/S0002-9904-1974-13612-8
5. Alexander Kuznetsov and Alexander Perry, "Derived categories of Gushel–Mukai varieties", Compositio Mathematica 154 (2018), no. 7, 1362–1406, DOI 10.1112/S0010437X18007091, arXiv:1605.06568, https://arxiv.org/abs/1605.06568
6. Jinhyung Park, "Some remarks on smooth projective varieties of small degree and codimension", Mathematische Zeitschrift 312 (2026), no. 3, article 74, DOI 10.1007/s00209-026-03965-1, arXiv:2405.11933, https://arxiv.org/abs/2405.11933