Curves on the product of two $K-$trivial surfaces

Abstract: We study curves on the product of two $K$-trivial surfaces. In the case of the product of two very general abelian surfaces $A_1\times A_2$, we prove that the minimal genus of a non-trivial curve on $A_1\times A_2$ is $6$.

• The paper establishes that the minimal genus for a non-trivial irreducible curve on a product of abelian surfaces is exactly 6, ruling out lower-genus cases.
• It employs explicit geometric constructions, deformation theory, and Hodge theory to derive sharp genus bounds and clarify moduli dimensions.
• The findings provide key insights into the birational geometry and moduli of abelian and K3 surfaces, linking measures of irrationality with Jacobian structures.

Curves on the Product of Two $K$-Trivial Surfaces

Introduction and Motivations

The paper "Curves on the product of two $K$-trivial surfaces" (2604.11799) undertakes a systematic study of the geometry of curves on products of $K$-trivial surfaces—primarily, very general abelian surfaces and $K3$ surfaces. The primary motivation is to compute invariants such as the minimal covering genus of correspondences in products of such varieties. This delivers sharp insight into the structure of moduli for abelian and $K3$ surfaces and their interplay via algebraic curves. The authors provide explicit genus bounds, clarify optimality, and connect these results to contemporary questions on measures of irrationality and Hodge-theoretic structures.

Summary of Main Results

Minimal Genus on $(A_1 \times A_2)$

Principal result: For two very general abelian surfaces $A_1$ and $A_2$, the minimal genus $g$ of a non-trivial irreducible algebraic curve $C \subset A_1 \times A_2$ with nonconstant projections to both factors is exactly $K$0. This is realized by the existence of a curve of genus $K$1 and proven optimal by showing that families of curves of genus $K$2 do not possess the required surjectivity in their Jacobians to $K$3. Hence,

$K$4

This value is strictly lower than the minimal genus in the case of a very general abelian fourfold, which reflects the special structure of the isogeny class of the product.

Minimal Genus on $K$5

An explicit construction similarly gives $K$6 when $K$7 is a very general $K$8 and $K$9 is a very general abelian surface. The lower bound is established by a parameter count and rigidity argument—curves of genus $K$0 cannot provide moving families covering both factors by general type and simplicity constraints.

Hodge-Theoretic Reformulation

The minimal genus for which a Jacobian can have $K$1 as a quotient is precisely $K$2. These results are reinterpreted via the language of correspondences and the covering genus $K$3 as introduced by Lazarsfeld and Martin [RobOli].

Methodologies and Proof Strategies

The proof employs a mix of:

• Explicit Geometric Constructions: Building fibered products of genus $K$4 curves and lifting hyperelliptic structures, imposing branch coincidence, to realize genus $K$5 curves with nontrivial projections to each factor.
• Deformation and Parameter Count: Using Hilbert schemes, Torelli loci, and the monodromy of families of Jacobians to bound the dimension of relevant moduli spaces.
• Hodge Theory and Prym Varieties: A careful analysis of the mixed Hodge structures arising from isogeny decompositions of Jacobians, applying the variational properties of the Torelli map and the study of multiplication maps on holomorphic differentials.
• Rigidity and Simplicity Arguments: Exclusion of nontrivial covers of genus $K$6 by demonstrating impossibility to independently vary curves and their maps into both abelian surfaces except in lower codimension.

Notable Technical Results

• The lower bound of $K$7 for the minimal genus is derived by analyzing the rank of the multiplication map from sections of tensor products of holomorphic differentials (related to the infinitesimal Torelli theorem on products and Prym loci).
• It is shown that for very general $K$8, any irreducible curve $K$9 with nonconstant projections must have Jacobian surjecting to $K3$0. Hence, any such genus $K3$1 curve would yield a $K3$2-dimensional locus in moduli of genus $K3$3 curves whose Jacobians split off two abelian surfaces—a dimension count invalidates this possibility for $K3$4.
• The Torelli locus in $K3$5 intersects $K3$6 along loci of dimension at most $K3$7, and the intersection is not of general type nor sufficiently ample to provide genus $K3$8 curve correspondences for very general products.

Theoretical and Practical Implications

These results quantify the complexity of correspondences between two $K3$9-trivial surfaces beyond classical birational invariants: the minimal genus of a non-product curve cannot be lower than $K3$0 on general products.

• Birational and Hodge-theoretic consequences: This establishes a sharp distinction between general abelian fourfolds and the special case of products of abelian surfaces, contributing to the understanding of measures of irrationality for higher-dimensional varieties [Olivier-Abelian, Nathan-Abelian] and extending invariant-theoretic methods to moduli of products.
• Connection with Moduli of Curves: The work builds upon and complements recent advances in understanding the geometry and parameterization of abelian varieties and their subvarieties, such as the complete solution for optimal covering genus in dimension up to $K3$1 via the Prym-Tyurin construction [engel2025optimalityprymtyurinconstructionmathcala6, AlexeevDonagiFarkasIzadiOrtega+2020+163+217].
• Classical and Motivic Geometry: By linking the existence and absence of low genus curves to isogeny factors in Jacobians and moduli dimensions, the paper deepens the interplay between algebraic, Hodge, and motivic structures.

Future Directions

Several questions naturally arise:

• Extension of these results to products of higher-dimensional abelian varieties and $K3$2-trivial varieties.
• The study of non-simple and special abelian surfaces, or more generally, products with prescribed endomorphism structure.
• The exploration of similar genus bounds for higher codimensional cycle classes and correspondence degrees in products of varieties with trivial or non-trivial canonical bundles.
• Analysis of the arithmetic analogs, such as the fields of definition of such covering families and their Galois/asymptotic properties.

Conclusion

The paper delivers a definitive result on the minimal geometric genus for nontrivial curves on products of two very general abelian surfaces, establishing it as $K3$3. The methodology hinges on a sophisticated interplay between deformation theory, Hodge structures, geometry of moduli spaces, and explicit constructions. The results refine our understanding of correspondences and measures of irrationality for abelian varieties, and will serve as a reference point for further study in the birational and arithmetic geometry of $K3$4-trivial and abelian-type varieties (2604.11799).