Obstructions to embedding singular curves in toric varieties

Technical Summary: Obstructions to Embedding Singular Curves in Toric Varieties

Problem Statement
The paper addresses the problem of embedding projective curves into toric varieties. While every smooth projective curve embeds into $\mathbb{P}^3$, singular curves face local obstructions: if a singular point $p$ has a Zariski tangent space of dimension $d$, the curve cannot embed into any smooth projective variety of dimension strictly less than $d$. A natural strategy to circumvent this is to embed the curve into a singular ambient variety, such as a weighted projective space or a general toric variety, where the tangent space at a singular point may be sufficiently large.

The central question posed is whether every projective curve can be embedded into some weighted projective threefold (Question 1.1). The authors investigate whether there exist reduced, irreducible curves that embed in $\mathbb{P}^d$ but fail to embed in any projective normal toric variety of dimension less than $d$.

Methodology
The authors employ a local-to-global strategy, focusing on the geometry of tangent cones at singular points.

1. Local Obstruction via Line Arrangements: The core of the argument involves analyzing "concurrent line arrangements" $C_\Gamma$, defined as the cone over a set of $m$ points $\Gamma$ in a hyperplane $H \subset \mathbb{P}^d$ with vertex $p$. The projectivized tangent cone at the vertex is isomorphic to the configuration $\Gamma$.
2. Dimension Counting and Moduli Spaces: The authors compare the dimension of the moduli space of such line arrangements against the dimension of families of tangent cones available in target varieties.
   • They establish that for a sufficiently large number of lines $m$, the moduli space of isomorphism classes of $C_\Gamma$ has dimension $m(d-1) - d^2 + 1$ (Corollary 3.3).
   • They analyze families of $\delta$-dimensional varieties (where $\delta < d$) and show that the tangent cones occurring in these families vary in families of bounded dimension.
   • By counting dimensions, they demonstrate that if $m$ is large enough, a "very general" arrangement $C_\Gamma$ cannot admit a vertex-preserving closed immersion into the tangent cone of any point in a variety of dimension $\delta < d$ (Theorem 3.6).
3. Construction of Singular Curves: To apply this local obstruction to actual curves, the authors construct a reduced, irreducible curve $C \subset \mathbb{P}^d$ with a unique singular point $p$ such that the completed tangent cone $TC_p C$ contains a prescribed arrangement $C_\Gamma$ (Proposition 4.1). This is achieved via a sequence of blow-ups and applications of Bertini's Theorem.
4. Distinction between Normal and Non-normal Toric Varieties: The authors distinguish between normal and non-normal toric varieties. They show that while normal toric varieties have only countably many isomorphism classes of tangent cones (allowing the dimension count to rule out embeddings), non-normal toric varieties can be constructed to contain any specific line arrangement (Theorem 3.11).

Key Results

• Main Theorem (Theorem 1.2 & Theorem 4.2): For every integer $d \geq 2$, there exists a reduced, irreducible projective curve $C_d \subset \mathbb{P}^d$ that does not embed in any projective normal toric variety of dimension less than $d$.
  • This result is generalized to show that for any countable collection of families of varieties with dimensions strictly smaller than $d$ and bounded base dimensions, there exists a curve that embeds in none of them.
• Corollary on Weighted Projective Spaces: Since weighted projective spaces are normal toric varieties, the main theorem implies that for every $d \geq 2$, there exists a curve in $\mathbb{P}^d$ that does not embed in any weighted projective space of dimension less than $d$. This provides a negative answer to Question 1.1 for $d=4$.
• Local vs. Global Obstructions: The paper clarifies that the obstruction is sensitive to the "normal" hypothesis.
  • Normal Case: The local obstruction (tangent cone dimension) is sufficient to prevent embedding into normal toric varieties of lower dimension.
  • Non-normal Case: Every concurrent line arrangement $C_\Gamma$ admits an embedding into a non-normal projective toric threefold (Theorem 3.11).
  • Linear Projections: Every $C_\Gamma$ can be embedded in a linear projection of a weighted projective plane $\mathbb{P}(1, 1, n)$ for large $n$, though these projections are generally not toric (Theorem 3.9).
• Global Obstructions for Smooth Curves: The authors note that even for smooth curves, global obstructions exist. For example, general smooth curves of genus $g > 10$ do not embed into cones over smooth rational curves (Remark 4.3), nor into any Hirzebruch surface.

Significance and Scope
The paper establishes that the class of projective normal toric varieties (and weighted projective spaces) is not "universal" for embedding projective curves, even when allowing for singular ambient spaces of lower dimension. The authors demonstrate that the local geometry of the tangent cone at a singular point imposes strict constraints on the ambient variety's dimension if the variety is required to be normal and toric.

The work highlights a fundamental difference between normal and non-normal toric varieties regarding embedding capabilities. While normal toric varieties are rigid enough to be obstructed by dimension counts of tangent cones, non-normal toric varieties are flexible enough to accommodate any finite arrangement of concurrent lines.

The paper concludes by posing open questions regarding the classification of curves that do admit embeddings into normal toric threefolds and the extension of these embedding problems to higher-dimensional smooth varieties, noting that the situation for surfaces is significantly more subtle than for curves.