[Paper Review] Dimension counts for singular rational curves via semigroups

Ethan Cotterill, Lia Feital+1 more|arXiv (Cornell University)|Nov 26, 2015

Algebraic Geometry and Number Theory22 references3 citations

TL;DR

This paper investigates the dimension of Severi-type varieties parameterizing singular rational curves in projective space by analyzing their value semigroups. Using $γ^*$-hyperelliptic semigroups to model cusp singularities, the authors construct explicit families of unicuspidal rational curves whose parameterizations impose fewer linear conditions than expected, proving that for $n \geq 8$, the Severi-type variety $M^n_{d,g}$ contains excess components not in the closure of $g$-nodal curves—thus demonstrating reducibility and refuting a natural codimension heuristic.

ABSTRACT

We study singular rational curves in projective space, deducing conditions on their parametrizations from the value semigroups $\sss$ of their singularities. In particular, we prove that a natural heuristic for the codimension of the space of nondegenerate rational curves of arithmetic genus $g>0$ and degree $d$ in $\mb{P}^n$, viewed as a subspace of all degree-$d$ rational curves in $\mb{P}^n$, holds whenever $g$ is small. On the other hand, we show that this heuristic fails in general, by exhibiting an infinite family of examples of Severi-type varieties of rational curves containing "excess" components of dimension strictly larger than the space of $g$-nodal rational curves.

Motivation & Objective

• To determine whether the codimension of the space of nondegenerate rational curves of arithmetic genus $g > 0$ and degree $d$ in $\mathbb{P}^n$ matches the expected dimension based on a nodal curve heuristic.
• To investigate whether this heuristic holds for unicuspidal rational curves, particularly when $n \geq 3$, by analyzing the role of value semigroups in singularity classification.
• To construct explicit examples of rational curves with $g = 3\gamma + 1$ genus whose parameterizations fail to impose $ (n-2)g $ independent conditions, contradicting the nodal heuristic.
• To demonstrate that Severi-type varieties $M^n_{d,g}$ are reducible for $n \geq 8$ by exhibiting excess components not contained in the closure of the $g$-nodal locus.

Proposed method

• The authors define $\gamma^*$-hyperelliptic semigroups as numerical semigroups satisfying two conditions: containing all even integers from 1 to $4\gamma - 2$, and including $4\gamma$.
• They construct $t$-power series parameterizations $f_i(t) = t^{2(\gamma + i)} + O(t^{2(\gamma + i)+1})$ for $i = 0, \dots, \gamma - 1$, with generic higher-order coefficients, to model unibranch singularities.
• The value semigroup $\mathrm{S}^*$ of the resulting cusp decomposes as $\mathrm{S}^* = \mathrm{S}^*_0 + \mathrm{S}^*_1$, where $\mathrm{S}^*_0 = \langle 2\gamma, \dots, 4\gamma - 2 \rangle$ and $\mathrm{S}^*_1 = \langle 4\gamma + 5, \dots, 6\gamma + 3 \rangle$, generating even and odd valuations respectively.
• The ramification conditions from the cusp are counted via a formula $r(f) = \frac{(5\gamma - 3)\gamma}{2} - 1$, which counts linear conditions on coefficients from vanishing partial derivatives.
• The genus of the semigroup is computed as $g(\mathrm{S}^*) = 3\gamma + 1$, derived from the disjoint union of gap sets $G_0 = \{2, 4, \dots, 2\gamma - 2\}$ and $G_1 = \{1, 3, \dots, 4\gamma + 3\}$.
• By comparing $r(f)$ to the expected codimension $(n-2)g(\mathrm{S}^*)$, they show $r(f) < (n-2)g(\mathrm{S}^*)$ for $n \geq 8$ and $\gamma \geq 8$, proving excess dimension.

Experimental results

Research questions

• RQ1 Does the codimension of the space of nondegenerate rational curves of genus $g > 0$ in $\mathbb{P}^n$ match the heuristic based on $g$-nodal curves for $n \geq 3$?
• RQ2 Can value semigroups of unibranch singularities be used to systematically classify and estimate the dimension of Severi-type varieties of rational curves?
• RQ3 Are there examples of unicuspidal rational curves in $\mathbb{P}^n$ with $n \geq 8$ whose parameterizations impose fewer than $ (n-2)g $ independent conditions, violating the nodal heuristic?
• RQ4 Do Severi-type varieties $M^n_{d,g}$ remain irreducible for $n \geq 8$, or do they contain excess components not in the closure of the $g$-nodal locus?
• RQ5 Can the construction of $\gamma^*$-hyperelliptic semigroups be generalized to yield more examples of reducible Severi-type varieties?

Key findings

• For $n \geq 8$, there exist Severi-type varieties $M^n_{d,g}$ with $g = 3\gamma + 1$ and $d \gg g$ that contain excess components not contained in the closure of the $g$-nodal rational curves locus.
• The value semigroup $\mathrm{S}^*$ of the constructed cusp singularities is $\gamma^*$-hyperelliptic, with genus $g(\mathrm{S}^*) = 3\gamma + 1$, derived from gap sets $G_0$ and $G_1$ of size $\gamma - 1$ and $2\gamma + 2$ respectively.
• The number of ramification conditions $r(f)$ imposed by the cusp is $\frac{(5\gamma - 3)\gamma}{2} - 1$, which is strictly less than $(n-2)g(\mathrm{S}^*)$ for $n \geq 8$ and $\gamma \geq 8$, violating the expected codimension heuristic.
• The Severi-type variety $M^n_{d,g}$ is reducible for $n \geq 8$, as the excess components from $\gamma^*$-hyperelliptic cusps are not in the closure of the $g$-nodal locus.
• The construction relies on $t$-power series $f_i(t) = t^{2(\gamma + i)} + O(t^{2(\gamma + i)+1})$ with generic higher-order coefficients, yielding a unibranch singularity with a well-defined, decomposable value semigroup.
• The result shows that the natural heuristic based on nodal curves fails in higher ambient dimensions, and that semigroup-based stratification is essential for accurate dimension estimates in the moduli of singular rational curves.