Eisenbud–Harris conjecture in Castelnuovo theory: high-genus curves on low-degree surfaces and the hyperplane-section supporting conjecture

Castelnuovo theory studies how large the genus of an algebraic curve of fixed degree in projective space can be, and what geometry is forced near the maximum. Eisenbud and Harris conjectured in 1982 that a nondegenerate reduced irreducible curve $X\subset\mathbf{P}^{r+1}$ of degree $n\ge 2r+3+2\alpha$ and genus $g>\pi_\alpha(n,r)\approx n^2/(2(r+\alpha))$ must lie on a surface of degree at most $r-1+\alpha$. Only $\alpha=1$ (Eisenbud–Harris) and $\alpha=2$ in ambient dimension at least 8 (Petrakiev, 2008, via Gröbner bases), plus bounded-degree partials for $\alpha=3,4$, are known. The operative core is the supporting conjecture on the general hyperplane section $\Gamma\subset\mathbf{P}^r$: if $\Gamma$ imposes $2r+1+\alpha$ conditions on quadrics ($0\le\alpha\le r-2$) and $n\ge 2r+3+2\alpha$ (or $n\ge 4r+1$ when $\alpha=r-2$), then $\Gamma$ should lie on a reduced irreducible curve of degree at most $r+\alpha$ that is a component of the intersection of the quadrics containing $\Gamma$; this is known for $\alpha\le 3$ with ambient-dimension restrictions and for arbitrary $\alpha\le r-3$ when $\Gamma$ lies on a two-dimensional rational normal scroll. Rathmann's 2016 preprint proves an infinitesimal deformation-theoretic version: a set of sufficiently many points on a linearly normal curve of genus $\alpha$ and degree $r+\alpha$ cannot deform with constant number of quadric conditions without the curve deforming along, in exactly the ranges predicted by the conjecture. Both conjectures remain open for $\alpha\ge 3$ in unrestricted degree, for $\alpha=2$ in low ambient dimension, and in the boundary case $\alpha=r-2$.

Background

Work over an algebraically closed field $K$ (Rathmann's preprint states its results in arbitrary characteristic). A curve $X\subset\mathbf{P}^{r+1}$ is nondegenerate if it is not contained in any hyperplane; its degree $n$ is the number of points in a general hyperplane section, and $g$ is its (arithmetic) genus. The central tool is the Hilbert function $h_\Gamma(l)$ of a general hyperplane section $\Gamma=X\cap H\subset H\simeq\mathbf{P}^r$: the rank of the evaluation map $H^0\mathscr{O}_{\mathbf{P}^r}(l)\to H^0\mathscr{O}_\Gamma(l)$, i.e. the number of conditions $\Gamma$ imposes on degree-$l$ forms. Points of a general hyperplane section of an irreducible curve are in uniform position (any two subsets of equal cardinality have the same Hilbert function), which by subadditivity gives $h_\Gamma(l)\ge\min(n,lr+1)$; in particular $h_\Gamma(2)\ge 2r+1$, so saying that $\Gamma$ imposes $2r+1+\alpha$ conditions on quadrics names the first $\alpha$ steps beyond this Castelnuovo minimum.

Castelnuovo's lemma says that if $n\ge 2r+3$ and $h_\Gamma(2)=2r+1$, then the quadrics through $\Gamma$ cut out a unique rational normal curve of degree $r$ through $\Gamma$ (a rational normal curve — the image of $\mathbf{P}^1$ under the complete linear system of degree $r$, the curve of minimal degree in $\mathbf{P}^r$). Eisenbud and Harris proved the next case, previously known to Fano: if $n\ge 2r+5$ and $h_\Gamma(2)=2r+2$, then $\Gamma$ lies on an elliptic normal curve of degree $r+1$ (a curve of genus one embedded by a complete linear system). These lemmas drive the genus bounds: Castelnuovo's bound $g\le\pi_0(n,r)=n^2/(2r)+O(n)$ is attained only by curves on surfaces of minimal degree $r$ in $\mathbf{P}^{r+1}$ (rational normal scrolls — surfaces swept out by a one-parameter family of lines joining two rational normal curves, or cones over them — together with the Veronese surface in $\mathbf{P}^5$, the image of $\mathbf{P}^2$ under the complete system of conics).

To interpolate toward lower genera, Eisenbud and Harris defined, for $1\le\alpha\le r$, thresholds $\pi_\alpha(n,r)$ by

$\lambda_\alpha=\Big\lfloor\frac{n-1}{r+\alpha}\Big\rfloor,\quad\epsilon_\alpha=n-1-\lambda_\alpha(r+\alpha),\quad\mu_\alpha=\max\Bigl(0,\Bigl\lfloor\frac{\alpha-r+1+\epsilon_\alpha}{2}\Bigr\rfloor\Bigr),$

$h_\alpha(l)=\begin{cases}l(r+\alpha)-\alpha+1 & 1\le l\le\lambda_\alpha,\\ n-\mu_\alpha & l=\lambda_\alpha+1,\\ n & l\ge\lambda_\alpha+2,\end{cases}\qquad\pi_\alpha(n,r)=\sum_{l\ge 1}\bigl(n-h_\alpha(l)\bigr)=\binom{\lambda_\alpha}{2}(r+\alpha)+\lambda_\alpha(\epsilon_\alpha+\alpha)+\mu_\alpha,$

so that $\pi_\alpha(n,r)=n^2/(2(r+\alpha))+O(n)$ (for $\alpha=0$ this is Castelnuovo's bound; the formula follows the standard convention, as printed in Petrakiev's account, translated to the $\mathbf{P}^{r+1}$ parametrization of Rathmann's preprint). They proved that for $n$ above an explicit but exponential bound $n_0(r)$ (e.g. $n_0(r)=2^{r+2}$ for $r\ge 7$) the inequality $g>\pi_\alpha(n,r)$ forces $X$ onto a surface of degree at most $r-1+\alpha$, proved the case $\alpha=1$ at the sharp threshold, and conjectured the general statement: the Eisenbud–Harris conjecture. Petrakiev (2008) settled $\alpha=2$ in ambient dimension at least 8, by a Gröbner-basis analysis of the degree-2 ideal of $\Gamma$, and proved bounded-degree partial results for $\alpha=3$ and $\alpha=4$.

All known proofs pass through a statement about point sets — the supporting conjecture (Rathmann's Conjecture 1.2; except for its boundary case (ii), noted and studied by Reid and later by Petrakiev): a general hyperplane section imposing $2r+1+\alpha$ conditions on quadrics, in the stated degree ranges, lies on a reduced irreducible curve of degree at most $r+\alpha$ which is a component of the intersection of the quadrics containing $\Gamma$. The passage from points to surfaces is standard: $g>\pi_\alpha(n,r)$ forces the Hilbert function of $\Gamma$ below the generic one, so $h_\Gamma(2)=2r+1+\beta$ with $\beta$ in range; the supporting conjecture places $\Gamma$ on a curve $D$ of degree at most $r+\beta$; a normality lemma (an irreducible curve of degree $r+\beta'$ carrying $\Gamma$ forces $h_\Gamma(l)\ge h_{\beta'}(l)$ for all $l$, the required normality automatic in the relevant range) together with the monotonicities among the $\pi_\beta$ rules out all degrees above $r-1+\alpha$; and, when $X$ is linearly normal (embedded by a complete linear system), every quadric through $\Gamma$ extends to a quadric through $X$, so the base locus of the quadrics through $X$ acquires a two-dimensional component of degree equal to $\deg D$ containing $X$; the general case reduces to the linearly normal one by projection and induction on $\alpha$. The numerical hypotheses are sharp (Reid; Eisenbud–Green–Harris): the case-(i) bound is insufficient at $\alpha=r-2$ — the $4r$ points cut out on a del Pezzo surface of degree $r$ in $\mathbf{P}^r$ by two general quadrics impose $2r+1+\alpha$ conditions on quadrics, but (for $r\ge 4$, where such a surface is cut out by quadrics) the intersection of those quadrics is the finite set itself and has no curve component — which is why the supporting conjecture needs the separate boundary case (ii) with the stronger bound $n\ge 4r+1$ (the example has $n=4r$). The supporting conjecture is the symmetric-position special case of the uniform-position conjectures of higher Castelnuovo theory (Eisenbud–Green–Harris, 1993), which are also open.

Rathmann's 2016 preprint contributes an infinitesimal version of the supporting conjecture. Fix a linearly normal curve $C\subset\mathbf{P}^r$ of genus $g=\alpha$ and degree $r+\alpha$, embedded by a complete linear system — the extremal curve type predicted by the conjecture (an elliptic normal curve for $\alpha=1$); the homogeneous ideal of such a curve is generated by quadrics (property $(N_1)$), and $C$ imposes exactly $2r+1+\alpha$ conditions on quadrics. Let $H(n,\alpha)$ parametrize sets of $n$ distinct points on such curves, and let $P(n,\alpha)$ be the determinantal locus, in the variety of $n$-point subsets of $\mathbf{P}^r$, where the quadric evaluation map has rank at most $2r+1+\alpha$. Rathmann proves that the natural map $F(n,\alpha):H(n,\alpha)\to P(n,\alpha)$ maps isomorphically onto a dense open subset of a component precisely in the ranges predicted by the supporting conjecture (Theorem 1.4), and is a local isomorphism for $n\ge 2r+\alpha+5$ (Theorem 1.5). Infinitesimally: a deformation of $\Gamma$ keeping the number of quadric conditions constant forces $C$ to deform along with $\Gamma$. The proof reduces this to the surjectivity of $H^0\mathscr{I}_C(2)\otimes H^0B\to H^0(N^\vee_{C/\mathbf{P}}(2)\otimes B)$, where $N^\vee$ is the conormal bundle and $B=\omega_C\otimes\mathscr{O}_C(\Gamma-2)$, then to the vanishing of $H^1$ of a bundle $P_L$ obtained as a quotient of a direct image on $C\times C\times C$ of a product of line bundles twisted by $-\Delta_{12}-\Delta_{13}-\Delta_{23}$ (the three diagonals), established by a filtration of the direct image. Hyperelliptic examples (Example 3.4 of the source) show the degree bounds in these theorems are sharp. The method has stated limits (source §4.1): it needs $C$ linearly normal with quadric-generated ideal, and the paper exhibits linearly normal $X$ whose predicted hyperplane-section curve $C$ is not linearly normal, and situations where the intersection of the quadrics containing $\Gamma$ is a surface rather than a curve — regimes where neither the classical sweeping argument nor the infinitesimal one applies directly.

Problem Statement

Prove or refute, in the generality stated, the following two linked conjectures of Castelnuovo theory over an algebraically closed field. Throughout, the degree of a projective curve or surface is the number of its intersection points with a general linear subspace of complementary dimension; nondegenerate means contained in no hyperplane; reduced and irreducible are part of the hypotheses; the genus is the arithmetic genus; and the degree bounds are sharp and may not be weakened.

(E) The Eisenbud–Harris conjecture (Harris 1982, with the collaboration of Eisenbud; Conjecture 1.1 of Rathmann's preprint). Let $X\subset\mathbf{P}^{r+1}$ be a nondegenerate reduced irreducible curve of degree $n$ and genus $g$, with $2\le r$, $1\le\alpha\le r$ and $n\ge 2r+3+2\alpha$. Define

$\lambda_\alpha=\Big\lfloor\frac{n-1}{r+\alpha}\Big\rfloor,\qquad\epsilon_\alpha=n-1-\lambda_\alpha(r+\alpha),\qquad\mu_\alpha=\max\Bigl(0,\Bigl\lfloor\frac{\alpha-r+1+\epsilon_\alpha}{2}\Bigr\rfloor\Bigr),$

$\pi_\alpha(n,r)=\binom{\lambda_\alpha}{2}(r+\alpha)+\lambda_\alpha(\epsilon_\alpha+\alpha)+\mu_\alpha,$

the Eisenbud–Harris threshold, of size $n^2/(2(r+\alpha))+O(n)$ for fixed $r$ and $\alpha$ (for $\alpha=0$ it is Castelnuovo's classical bound). If $g>\pi_\alpha(n,r)$, then $X$ lies on a surface of degree at most $r-1+\alpha$.

(S) The supporting conjecture on hyperplane sections (Conjecture 1.2 of Rathmann's preprint; in essence Reid, and Petrakiev). Let $\Gamma\subset\mathbf{P}^r$ be the section of a nondegenerate reduced irreducible curve $X\subset\mathbf{P}^{r+1}$ of degree $n$ by a general hyperplane (so that $\Gamma$ consists of $n$ distinct points in uniform position: any two subsets of equal cardinality have the same Hilbert function), and suppose $\Gamma$ imposes $2r+1+\alpha$ conditions on quadrics for some $0\le\alpha\le r-2$ — that is, the rank of the evaluation map $H^0\mathscr{O}_{\mathbf{P}^r}(2)\to H^0\mathscr{O}_\Gamma(2)$ on quadrics equals $2r+1+\alpha$. If (i) $\alpha\le r-3$ and $n\ge 2r+3+2\alpha$, or (ii) $\alpha=r-2$ and $n\ge 4r+1$, then $\Gamma$ lies on a reduced irreducible curve $C$ of degree at most $r+\alpha$, and $C$ is a component of the intersection of the quadrics containing $\Gamma$.

The two statements are linked: (S) for parameter $\alpha+1$, combined with the Eisenbud–Harris Hilbert-function analysis (subadditivity, the normality lemma for the containing curve, the monotonicities of the $\pi_\beta$, and the reduction to the linearly normal case by projection and induction), implies (E) with parameter $\alpha$; this is the route by which all known cases were proved ($\alpha=0,1$ classically, $\alpha=2$ by Petrakiev in ambient dimension at least 8, with bounded-degree partials for $\alpha=3,4$). A complete answer settles both conjectures in their full stated ranges: a proof of (E) for every $r\ge 2$, every $1\le\alpha\le r$ and every $n\ge 2r+3+2\alpha$, together with a proof of (S) for every $0\le\alpha\le r-2$ including the boundary case (ii), or explicit counterexamples inside the stated windows resolving each conjecture. By the linkage above, a proof of (S) over its full range establishes (E) only for $\alpha\le r-3$, leaving the band $\alpha\in\{r-2,r-1,r\}$ of (E) to be settled separately, and a counterexample to one conjecture does not resolve the other. Results that add hypotheses (linear normality of $X$, stronger position assumptions on the points, characteristic zero, bounded degree) or cover only some values of $\alpha$ or $r$ are partial progress toward, not a solution of, the problem.

References

1. J. Rathmann, An infinitesimal approach to a conjecture of Eisenbud and Harris, preprint, arXiv:1604.06069 (2016), https://arxiv.org/abs/1604.06069 (unpublished).
2. J. Harris, with the collaboration of D. Eisenbud, Curves in projective space, Séminaire de Mathématiques Supérieures 85, Presses de l'Université de Montréal, Montréal, 1982 (zbMATH 3805712, https://zbmath.org/3805712).
3. I. Petrakiev, Castelnuovo theory via Gröbner bases, Journal für die reine und angewandte Mathematik 619 (2008), 49–73, doi:10.1515/crelle.2008.040, https://doi.org/10.1515/crelle.2008.040; preprint version: A step in Castelnuovo theory via Gröbner bases, arXiv:math/0604517, https://arxiv.org/abs/math/0604517.
4. M. Reid, Quadrics through a canonical surface, in: Algebraic geometry (L'Aquila, 1988), Lecture Notes in Mathematics 1417, Springer, Berlin, 1990, pp. 191–213 (zbMATH 4163065, https://zbmath.org/4163065).
5. D. Eisenbud, M. Green, J. Harris, Higher Castelnuovo theory, Astérisque 218 (1993), 187–202 (Journées de Géométrie Algébrique d'Orsay, 1992; zbMATH 592836, https://zbmath.org/592836).
6. G. Fano, Sopra le curve di dato ordine e dei massimi generi in uno spazio qualunque, Memorie della Reale Accademia delle Scienze di Torino, serie 2, 44 (1894), 335–382 (zbMATH 2683130, https://zbmath.org/2683130).
7. E. Arbarello, A. Bruno, E. Sernesi, On hyperplane sections of K3 surfaces, Algebraic Geometry 4 (2017), no. 5, 562–596, doi:10.14231/ag-2017-028, https://doi.org/10.14231/ag-2017-028; preprint arXiv:1507.05002, https://arxiv.org/abs/1507.05002 (v1 titled On two conjectures of J. Wahl).
8. J. Wahl, On cohomology of the square of an ideal sheaf, Journal of Algebraic Geometry 6 (1997), no. 3, 481–511 (zbMATH 1036370, https://zbmath.org/1036370).
9. M. Green, R. Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Inventiones Mathematicae 83 (1986), 73–90, doi:10.1007/bf01388754, https://doi.org/10.1007/bf01388754.
10. I. Coskun, G. Smith, Very free rational curves in Fano varieties, Journal of Algebra 611 (2022), 246–264, doi:10.1016/j.jalgebra.2022.08.009, https://doi.org/10.1016/j.jalgebra.2022.08.009; preprint arXiv:2106.01991, https://arxiv.org/abs/2106.01991.
11. Z. Ran, Balanced rational curves and minimal rational connectedness of Fano hypersurfaces, preprint, arXiv:2008.01235 (2020), https://arxiv.org/abs/2008.01235.
12. C. Ciliberto, T. Dedieu, Extensions of curves with high degree with respect to the genus, Épijournal de Géométrie Algébrique, special volume in honour of Claire Voisin (2024), paper no. 16, doi:10.46298/epiga.2024.11202, https://doi.org/10.46298/epiga.2024.11202; preprint arXiv:2304.01851, https://arxiv.org/abs/2304.01851.
13. C. Ciliberto, T. Dedieu, E. Sernesi, Wahl maps and extensions of canonical curves and K3 surfaces, Journal für die reine und angewandte Mathematik 761 (2020), 219–245, doi:10.1515/crelle-2018-0016, https://doi.org/10.1515/crelle-2018-0016; preprint arXiv:1802.04618, https://arxiv.org/abs/1802.04618.