Curves in the Complex Projective Plane

• Curves in the complex projective plane are defined as the zero loci of homogeneous polynomials, exhibiting rich algebraic, topological, and differential properties.
• They showcase explicit normal forms, singularity classifications, and intrinsic invariants like the j-invariant that facilitate moduli space mapping and enumerative counts.
• Their study bridges algebraic geometry with applications in symplectic topology, hyperbolicity criteria, and combinatorial syzygy theory.

Curves in the complex projective plane are central objects in algebraic geometry, exhibiting a rich interplay of topological, differential, and algebraic properties. The study encompasses their classification, intrinsic and extrinsic invariants, singularity theory, group structures, moduli spaces, and deep connections to enumerative geometry and mathematical physics.

1. Fundamental Definitions and Normal Forms

A curve in the complex projective plane, denoted $\mathbb{CP}^2$, is the vanishing locus $$C = \{ [x : y : z] \in \mathbb{CP}^2 : F(x, y, z) = 0 \}$$ of a homogeneous polynomial $F$ of degree $d$. Points in $\mathbb{CP}^2$ are described by homogeneous coordinates $[x : y : z]$, up to multiplication by a nonzero complex scalar. The local structure is given by affine charts; for example, in the chart $z \neq 0$ (set $z=1$), $C$ is defined by $F(x, y, 1) = 0$.

Normal forms play a critical role in the classification and study of special classes of curves:

• Hesse normal form for cubics: Every smooth cubic is projectively equivalent to $x^3+y^3+z^3 - 3kxyz = 0$, with $k \in \mathbb{C}$, $k^3 \neq 1$. Singularities in this pencil occur for $k^3=1$ or $k=\infty$.
• Weierstrass normal form for cubics: Any smooth cubic with a chosen flex point can be written as $y^2z = x^3 + a x z^2 + b z^3$. Under projective changes fixing $[0:1:0]$, the coefficients transform as $x \mapsto t^2 x$, $y \mapsto t^3 y$, with $a \mapsto t^4 a$, $b \mapsto t^6 b$, leading to a complete invariant $J = 4a^3/(4a^3 + 27b^2)$. The associated $j$-invariant parametrizes the moduli of smooth cubics up to projective equivalence. The relation $j = 1728\cdot J$ maps the moduli space to $\mathbb{C} \cup \{\infty\}$ (Bonifant et al., 2016).

2. Singularities, Rational Curves, and Cusp Structure

Singularities are detected by vanishing of the gradient: a point $p$ is singular if all partial derivatives of $F$ vanish at $p$. The discriminant $\Delta$ of $F$ encodes the locus of singular curves; in the Weierstrass form, $\Delta = -16 (4a^3+27b^2)$.

For rational curves (homeomorphic to $\mathbb{P}^1$), the genus-degree formula gives the constraint $\sum_p \delta_p = \frac{(d-1)(d-2)}{2}$, where $\delta_p$ is the local $\delta$-invariant at a singularity $p$. The maximal number of singular points for a rational plane curve is thus bounded by this sum. In particular, a rational curve in $\mathbb{CP}^2$ has at most four singular points; equality is achieved uniquely (up to projective equivalence) for a quintic with four cusps, each with specified local analytic type. Explicit parametrizations and equations are provided for such extremal curves (Koras et al., 2019).

Cuspidal curves—those whose singularities are all analytically irreducible—admit classification via Newton pairs and multiplicity sequences at each cusp. For bicuspidal rational curves, explicit families can be constructed using iterated Cremona transformations, their singularity types determined and tabulated (Bodnár, 2016). A major classification achievement is the complete description of rigid unicuspidal rational curves of minimal degree with prescribed $(p,q)$ cusp: such a curve of degree $d$ exists if and only if $p+q=3d$ and either $(p,q)$ lies in a discrete Fibonacci family with $p/q<\tau^4$ or $p/q>\tau^4$, where $\tau=(1+\sqrt{5})/2$ and $\tau^4\approx6.854$ (McDuff et al., 30 Nov 2024).

3. Intrinsic and Extrinsic Invariants

Projective and Differential Invariants

Key algebraic invariants include the $j$-invariant for cubics, the genus $g$, the arithmetic and geometric genus of a curve, and the configuration of flex points, such as the classical $9_4$ arrangement for cubics. Holomorphic differentials, such as $dw = dx/y$ in Weierstrass form, provide the universal covering mapping to $\mathbb{C}/\Lambda$ with lattice $\Lambda$ encoding the conformal class of the curve (Bonifant et al., 2016).

The Green–Griffiths jet bundle offers a finely-graded structure of higher-order holomorphic differential operators, with explicit extrinsic generators achieved for smooth curves of degree $d \ge \kappa+3$. This yields algebraic obstructions to the existence of non-constant entire curves into $C$, and ultimately, hyperbolicity for curves of sufficiently high degree ($d \ge 5$) (Merker, 2014).

The Normal Map and Duality

The normal map $\nu: C \to (\mathbb{CP}^2)^\vee$ sends a point $p$ to the normal line at $p$ in the dual projective plane. For smooth curves in characteristic zero, this map is birational onto its image. For degrees $d>4$, the normal-line locus uniquely determines $C$. In positive characteristic, pathologies arise—strange curves become possible where the Gauss map is inseparable, causing distinct curves to share the same locus of normal lines (Ballico et al., 2021).

In the context of applied algebraic geometry, the locus of bottlenecks—the set of lines normal to $C$ at two distinct points—is generically finite for smooth curves in characteristic zero, with degree computable in terms of $d$: $\deg B(C) = d(d-2)(d-3)(d+3)$.

4. Special Families: Free, Nearly Free, and Unexpected Curves

A plane curve $C: f=0$ is defined as free if its module of Jacobian syzygies $AR(f)$ is free of rank two, equivalently if the sheaf of logarithmic vector fields splits as a direct sum of line bundles. The exponents $(d_1, d_2)$ (degrees of generators) govern key numerical invariants: $\tau(C) = (d-1)^2 - d_1 d_2,$ where $\tau(C)$ is the total Tjurina number. Nearly free and almost free curves generalize this behavior, characterized respectively by certain mild defects in the local cohomology module.

Line arrangements offer a wealth of explicit examples: unions of concurrent lines, supersolvable arrangements, and deformations via addition or deletion yield (nearly) free curves with controlled syzygy data. Certain rational cuspidal curves, notably maximizing ADE curves, also fall into the free or nearly free categories. Terao's conjecture, which posits the combinatorial determinacy of freeness for line arrangements, remains open (Dimca, 2023, Malara et al., 2020).

5. Enumerative Geometry, Moduli, and Holomorphic Curves

The moduli space of smooth complex cubics is bijective with $\mathbb{CP}^1$ via the $j$-invariant. Automorphism groups of these curves jump at special $j$-values (notably $j=0,\, 1728$), corresponding to enhanced symmetries of the lattice of periods. Projective duality, normal maps, and combinatorial arrangements (e.g. flex points, Hessian curves) yield rich geometrical content with bijective or finite-to-one correspondences.

Recent advances have focused on counting and classifying rational curves with prescribed singularities, especially unicuspidal curves with $(p,q)$-cusps. The complete characterization of when such rigid curves exist—based on an index-zero condition $p+q=3d$, Fibonacci-type discrete families, and the golden-ratio threshold—has significant implications for symplectic topology, with the associated counts assembling into the so-called ellipsoidal superpotential (McDuff et al., 30 Nov 2024, Siegel, 23 Apr 2024). The latter is governed combinatorially by explicit weighted sums over rooted trees, manifesting monotonicity and nonvanishing properties crucial to applications in symplectic embedding problems.

Holomorphic curves from $\mathbb{C}$ to $\mathbb{CP}^2$ avoiding various linear obstructions elucidate hyperbolicity phenomena: while omitting five projective lines forces constancy (Green's theorem), weakening to four complex hyperplanes plus a real subspace allows the existence of non-constant entire curves (Haggui et al., 2019).

6. Applications and Connected Research Directions

The algebraic–geometric study of projective plane curves interfaces with a multitude of modern research directions:

• Symplectic Geometry: The existence and count of rigid unicuspidal rational curves directly yield symplectic embedding obstructions for four-dimensional ellipsoids into balls, with the "Fibonacci staircase" and phase transitions tracked by explicit capacity functions (McDuff et al., 30 Nov 2024).
• Value Distribution and Hyperbolicity: The existence of global jet differentials for high-degree curves obstructs entire holomorphic maps, contributing to Kobayashi hyperbolicity criteria (Merker, 2014).
• Bottleneck Theory and Medial Axes: The normal map and its degenerate loci have direct applications to the geometry of offsets and medial axis computations in applied sciences (Ballico et al., 2021).
• Combinatorics and Syzygy Theory: Terao's conjecture and the structure of (nearly) free divisors motivate interplay between combinatorial arrangements and syzygy theory, with consequences for moduli and deformation theory (Dimca, 2023, Malara et al., 2020).
• Singularity and Surface Link Theory: Classification of rational cuspidal curves is deeply tied to the study of superisolated singularities and invariants of links in low-dimensional topology (Koras et al., 2019).

The fundamental classification, explicit construction, and enumeration of curves in the complex projective plane continue to drive advances at the intersection of algebraic geometry, topology, and neighboring disciplines.