A result on the generic Picard number of surfaces in fake weighted projective 3-spaces

1. Problem Statement and Context

The paper addresses a specific question in algebraic geometry regarding the Picard number ($\rho$) of generic hypersurfaces in "fake" weighted projective 3-spaces.

• The Setting: Let $\Delta$ be a 3-dimensional lattice simplex. Let $P_\Delta$ be the projective toric variety associated with the normal fan of $\Delta$. If $\Delta$ is not the standard simplex, $P_\Delta$ is a "fake weighted projective 3-space." Let $f$ be a nondegenerate Laurent polynomial with Newton polytope $\Delta$, and let $Z_{\Delta, f}$ be the closure of the zero set of $f$ in $P_\Delta$.
• The Conjecture: It is generally conjectured (analogous to the Noether-Lefschetz theorem) that for a "very general" (generic) choice of $f$, the Picard number of the surface $Z_{\Delta, f}$ is minimal, i.e., $\rho(Z_{\Delta, f}) = 1$. This implies that all algebraic cycles on the surface are proportional to the canonical divisor (or the restriction of the ambient space's divisor).
• The Goal: The author seeks to identify specific conditions on the Newton polytope $\Delta$ under which this conjecture fails, specifically proving that $\rho(Z_{\Delta, f}) > 1$ for certain generic surfaces. The surfaces in question are of general type.

2. Methodology

The paper employs a strategy based on degeneration theory and Hodge theory. Instead of analyzing the generic fiber directly, the author studies a degeneration of the surface along an edge of the Newton polytope.

A. Geometric Setup and Subdivision

• Assumptions: The simplex $\Delta$ must satisfy:
  1. $l^*(\Delta) > 0$ (ensuring the geometric genus $p_g > 0$).
  2. $\dim F(\Delta) \in \{2, 3\}$ (ensuring the surface is of general type, i.e., $K^2 > 0$).
  3. Existence of an edge $e \subset \Delta$ such that $l^*(e) > 0$ (where $l^*(F)$ denotes the number of lattice points in the relative interior of $F$).
• Subdivision: The edge $e$ is used to subdivide $\Delta$ into $r$ smaller simplices $\Delta_1, \dots, \Delta_r$ along the edge. The intersection of adjacent simplices $\Delta_i \cap \Delta_{i+1}$ forms a 2-dimensional face.

B. Semistable Degeneration

The author constructs a one-parameter family of surfaces $X_t$ over a unit disc $B$, where $X_t$ corresponds to the surface defined by $f$ for $t \neq 0$, and the central fiber $X_0$ is a singular surface arising from the subdivision.

• Resolution: Through a base change and birational maps, a semistable degeneration is achieved. The central fiber $X_0$ consists of a union of smooth surfaces $V_i$ (closures of hypersurfaces in the toric varieties of the sub-simplices) intersecting transversally along double curves $C_{i, i+1}$.
• Geometric Genus Tracking: A key formula relates the geometric genus of the generic fiber to the components of the central fiber:
  $$p_g(X_t) = \sum_{i=1}^r p_g(V_i) + \sum_{i=1}^{r-1} g(C_{i, i+1})$$
  Using lattice point counts, this translates to:
  $$l^*(\Delta) = \sum_{i=1}^r l^*(\Delta_i) + \sum_{i=1}^{r-1} l^*(\Delta_i \cap \Delta_{i+1})$$

C. Cohomological Analysis (Clemens-Schmid Sequence)

The core of the proof relies on the Clemens-Schmid exact sequence and the theory of vanishing cohomology.

• Monodromy: The monodromy operator $T$ on $H^2(X_t, \mathbb{Q})$ is analyzed. The operator $N = \log(T)$ measures the "vanishing" part of the cohomology.
• Vanishing Classes: The space of vanishing cohomology classes is defined as $H^2(X_t, \mathbb{Q})_{van} \cong H^2(X_t, \mathbb{Q}) / \ker(N)$.
• Orthogonality: These vanishing classes are orthogonal to the cohomology classes coming from the central fiber (the kernel of $N$) with respect to the intersection product.
• Hodge Decomposition: The author analyzes the position of these vanishing classes within the Hodge decomposition $H^2(X_t, \mathbb{C}) = H^{2,0} \oplus H^{1,1} \oplus H^{0,2}$.

3. Key Contributions and Results

The paper establishes a precise criterion for the Picard number to exceed 1.

The Main Theorem (Theorem 1.2 / Theorem 2.10)

Let $\Delta$ be a 3-dimensional simplex satisfying the conditions above, with a subdivision $\Delta = \bigcup \Delta_i$ along an edge $e$. If the following inequality holds:
$$\sum_{i=1}^{r-1} l^*(\Delta_i \cap \Delta_{i+1}) > \sum_{i=1}^{r} l^*(\Delta_i)$$
Then, for a sufficiently nondegenerate $f$, the Picard number of the surface is strictly greater than 1:
$$\rho(Z_{\Delta, f}) > 1$$

Mechanism of the Proof

1. Counting Vanishing Classes: The number of vanishing cohomology classes in the semistable degeneration is calculated as $2 \cdot \sum_{i=1}^{r-1} l^*(\Delta_i \cap \Delta_{i+1})$.
2. Dimensional Argument: The space $H^{2,0}(X_t) \oplus H^{0,2}(X_t)$ has dimension $2 \cdot p_g(X_t) = 2 \cdot \sum l^*(\Delta_i)$.
3. Existence of Rational (1,1) Class: If the number of vanishing classes (numerator) exceeds the dimension of the $(2,0)+(0,2)$ part (denominator), there must exist a vanishing cohomology class $V$ that is not in $H^{2,0} \oplus H^{0,2}$.
4. Rationality: Since $V$ is a rational class and orthogonal to the kernel of $N$ (which contains the canonical divisor $K_{X_t}$), its projection $w$ onto $H^{1,1}(X_t)$ is a rational Picard class.
5. Independence: Because $w$ is orthogonal to $K_{X_t}$ and $K_{X_t}^2 > 0$ (general type), $w$ cannot be proportional to $K_{X_t}$. Thus, the Picard group is generated by at least two independent classes.

Counter-Example to the Conjecture

The paper demonstrates that the conjecture $\rho=1$ is false for generic surfaces in fake weighted projective spaces if the Newton polytope satisfies the specific lattice point inequality derived above. This highlights that the geometry of the Newton polytope (specifically the distribution of interior lattice points relative to the subdivision) dictates the Picard number.

4. Significance

• Refinement of Noether-Lefschetz Theory: While the classical Noether-Lefschetz theorem states that generic surfaces in $\mathbb{P}^3$ have $\rho=1$, this work shows that in the context of fake weighted projective spaces (toric varieties with singularities), the geometry of the Newton polytope can force the Picard number to be higher even for generic members.
• Methodological Insight: The paper provides a concrete application of degeneration techniques (semistable reduction) and mixed Hodge structures to compute invariants of generic algebraic varieties. It bridges combinatorial data (lattice points in polytopes) with complex geometric invariants (Picard numbers).
• Constraints on General Type: The result emphasizes the importance of the condition $K^2 > 0$ (general type). The author notes in Remark 2.11 that for K3 surfaces (where $K^2=0$), similar degenerations do not necessarily yield $\rho > 1$ for generic members, as the Hodge structure behaves differently.
• Future Directions: The work suggests that the "generic" behavior of Picard numbers in toric varieties is more nuanced than in standard projective space, depending heavily on the combinatorial properties of the defining polytope.

In summary, Giesler proves that for specific classes of surfaces in fake weighted projective spaces, the generic Picard number is strictly greater than 1, providing a combinatorial criterion based on the subdivision of the Newton polytope and the count of interior lattice points.