Technical Summary: Curves Contained in a Quartic Determinantal Surface Containing a Line

Problem Statement

The paper investigates the geometry of curves contained in a very general smooth quartic surface $X \subseteq \mathbb{P}^3$ that contains a line $L$. While the geometry of curves on K3 surfaces (of which smooth quartics are the simplest examples) is a central theme in algebraic geometry, previous work by the authors ([CV26]) had already provided a complete classification of arithmetically Cohen-Macaulay (ACM) curves on such surfaces. Specifically, for a surface where $\text{Pic}(X)$ is generated by the hyperplane class $H$ and the line class $L$ (with intersection numbers $H^2=4, H\cdot L=1, L^2=-2$), a curve of class $aH + bL$ is ACM if and only if $b \in \{0, 1, -1\}$.

The primary problem addressed in this work is to extend the analysis beyond the ACM locus to study all curves contained in $X$, regardless of whether they are ACM. The authors aim to determine:

1. Which linear systems on $X$ contain smooth irreducible curves.
2. The behavior of these curves within the Hilbert scheme $\mathcal{H}_{d,g}$ of curves in $\mathbb{P}^3$, specifically determining if the general member is a smooth or singular point.
3. The cohomological behavior of these curves via the computation of their Rao functions.

Methodology

The authors employ a combination of classical algebraic geometry techniques, cohomological calculations, and computational algebra:

• Linear System Analysis: To distinguish between classes containing smooth irreducible curves and those where the general member is necessarily reducible, the authors apply a criterion by Kleppe [Kle17]. They analyze the intersection numbers of divisors with the line $L$ and the residual class $H-L$.
• Cohomological Computations: The paper utilizes long exact sequences in cohomology associated with short exact sequences of sheaves (e.g., relating the structure sheaf of a curve to that of the surface and the line). This allows for the calculation of $h^0$ and $h^1$ dimensions, which are crucial for determining the dimension of linear systems and the smoothness of points in the Hilbert scheme.
• Biliaison Theory: The authors utilize the concept of elementary biliaison (specifically $(4,1)$-biliaison) to relate curves in different linear systems. By linking curves to simpler known curves (such as $F_m \in |mL|$ or $G_m \in |mH - mL|$), they derive minimal free resolutions and compute Rao functions inductively.
• Computational Verification: For cases where theoretical decomposition of reducible divisors is complex, the authors use the software Macaulay2 to compute specific cohomological dimensions ($h^0(C, \mathcal{N}_C)$ and $h^1(C, \mathcal{N}_C)$) and verify the smoothness of specific curve families.
• Incidence Varieties: To determine the dimension of components in the Hilbert scheme for specific singular cases, the authors analyze incidence varieties involving the family of determinantal quartic surfaces.

Key Contributions and Results

1. Existence of Smooth Irreducible Curves

The authors classify the linear systems $|(k+m)H - mL|$ and $|kH + mL|$ based on the existence of smooth irreducible members:

• Smooth Existence: Smooth irreducible curves exist in $|(k+m)H - mL|$ for all $k, m \geq 1$. They also exist in $|kH + mL|$ provided $m \geq 1$ and $k \geq 2m$.
• Reducibility:
  • For $|kH + mL|$ with $1 \leq k \leq 2m-2$, the linear system contains no smooth irreducible curves.
  • For $|(2m-1)H + mL|$, the general member is reducible, consisting of the line $L$ union a smooth curve in $|(2m-1)H + (m-1)L|$.
  • For $|mH - mL|$, the general member is the union of $m$ plane cubics contained in $X$.

2. Hilbert Scheme Behavior

The paper analyzes the position of these curves in the Hilbert scheme $\mathcal{H}_{d,g}$:

• Class $|(k+m)H - mL|$:
  • If $k \geq 5$ and $m \geq 1$, the general member is a smooth point of a generically smooth component of dimension $2k^2 + mk + 34$.
  • If $k \in \{1, 2, 3\}$ and $m \geq 1$, the general member is a smooth point of a component of dimension $4(4k+3m)$.
  • If $k=4$ and $m \geq 2$, the general member is a singular point of the Hilbert scheme. The authors prove this by showing the dimension of the component containing these curves is strictly less than the expected dimension derived from the normal bundle cohomology.
• Class $|kH + mL|$:
  • If $k > 2m + 2$ and $m \geq 1$, the general member is a smooth point of a component with the expected dimension.
  • If $k \in \{2m, 2m+1, 2m+2\}$, the general member lies in a non-reduced component of the Hilbert scheme.
  • For remaining cases ($m \geq 2$ and $1 \leq k \leq 2m-1$), the divisors are reducible. The authors provide specific geometric descriptions for small cases (e.g., unions of lines and plane curves) and use Macaulay2 to compute normal bundle cohomology, though smoothness is not always conclusively determined for all sub-cases.

3. Rao Functions

The authors compute the Rao function $\rho_C(n) = \dim H^1(\mathbb{P}^3, \mathcal{I}_C(n))$ for any curve $C$ on $X$.

• They first derive explicit formulas for the Rao function of curves $F_m$ (defined by the ideal $I_{F_m} = I_L^m + \langle F \rangle$) for $m \geq 3$.
• Using the theory of biliaison, they establish a general formula relating the Rao function of any curve $C$ in the classes $|(k+m)H - mL|$ or $|kH + mL|$ to that of $F_m$:
  $$\rho_C(n) = \rho_{F_m}(m + k - n)$$
• Explicit piecewise formulas are provided for $m=3, 4$, and the general case $m \geq 5$.

Significance and Scope

The paper claims to provide a comprehensive study of the curves on a very general quartic surface containing a line, moving beyond the restricted scope of ACM curves. The significance lies in:

1. Completing the Classification: By determining exactly which linear systems yield smooth curves and characterizing the reducible cases, the work fills the gap left by the previous ACM classification.
2. Hilbert Scheme Stratification: The identification of specific classes where the general curve is a singular point or lies in a non-reduced component contributes to the understanding of the local geometry of the Hilbert scheme of space curves.
3. Cohomological Data: The explicit computation of Rao functions and minimal free resolutions provides concrete data for the postulate character and the stratification of the Hilbert scheme by schemes with constant cohomology.

The authors maintain a modest tone, acknowledging that for certain reducible cases (specifically $kH + mL$ with small $k$), the decomposition of the general member is not fully known theoretically, and their insights rely on computational examples. They do not propose new applications but rather deepen the theoretical understanding of the geometry of curves on this specific class of K3 surfaces.