Rational Curves in Projective Toric Varieties

TL;DR

The paper develops a combinatorial framework of degree-$d$ Cayley structures to classify and construct embedded rational curves in projective toric varieties $X_\mathcal{A}$. It builds families $C_{\pi,\mathbf{f}}$ from Cayley data, proves a bijection between maximal smooth primitive degree-$d$ Cayley structures and irreducible components of $\mathrm{Hilb}_{dm+1}(X_\mathcal{A})$ with smooth general elements, and describes the torus-orbit closures in $\mathrm{Chow}_d(X_\mathcal{A})$ via a fan $\Sigma_\pi$. The work also analyzes degenerations, stabilizers, and singularities (cusps/nodes), and provides explicit results for conics, including a comparison with matroid polytopes, yielding tools for counting and understanding rational curves in toric settings. These contributions connect the image geometry of rational curves to combinatorial data, enabling systematic degeneration analysis and a structural view of the Hilbert and Chow landscapes for toric varieties.

Abstract

We study embedded rational curves in projective toric varieties. Generalizing results of the first author and Zotine for the case of lines, we show that any degree $d$ rational curve in a toric variety $X$ can be constructed from a special affine-linear map called a degree $d$ Cayley structure. We characterize when the curves coming from a degree $d$ Cayley structure are smooth and have degree $d$. We use this to establish a bijection between the set of irreducible components of the Hilbert scheme whose general element is a smooth degree $d$ curve, and so-called maximal smooth Cayley structures. Furthermore, we describe the normalization of the torus orbit closure of such rational curves in the Chow variety, and give partial results for the orbit closures in the Hilbert scheme.

Figures (11)

• Figure 1: The set $\mathcal{A}$ for a Fano threefold and the corresponding normal fan (Example \ref{['ex:fano']}).
• Figure 2: Fans for components of the Hilbert scheme of conics (Example \ref{['ex:fano']}).
• Figure 3: Three Cayley structures for a Fano threefold (Example \ref{['ex:cayley']}). On top, elements with common image under the Cayley structure are grouped by the dashed lines. On the bottom, the Cayley structure is defined only on the highlighted face. Elements are labeled by their images under the Cayley structure.
• Figure 4: Example images of imprimitive, cuspidal, and nodal Cayley structures with $d=3$, $\ell=2$ (Definitions \ref{['defn:primitive']}, \ref{['defn:cuspidal']}, \ref{['defn:nodal']}).
• Figure 5: A non-maximal Cayley structure (Example \ref{['ex:fanohilb']})

Theorems & Definitions (82)

• Theorem 1.1: See Corollary \ref{['cor:main']}
• Theorem 1.2: See Theorem \ref{['thm:fan']}
• Definition 2.2.1
• Definition 2.2.2
• Definition 2.2.3
• Remark 2.2.4
• Definition 3.1.1: Rational curve from a weak Cayley structure
• Proposition 3.1.2
• proof
• Lemma 3.1.3