[Paper Review] The Canonical Model of a Singular Curve

Steven L. Kleiman, Renato Vidal Martins|ArXiv.org|Mar 23, 2008

Algebraic Geometry and Number Theory14 references4 citations

TL;DR

This paper provides modern, refined proofs of Rosenlicht's foundational results on the canonical model $C'$ of a singular integral curve $C$ of genus $g \geq 2$. It establishes that $C$ and $C'$ are birationally equivalent if and only if $C$ is nonhyperelliptic, and shows that in this case, $C'$ is isomorphic to the blowup of $C$ with respect to its canonical sheaf $\omega$. The key contribution is a complete characterization of when $C'$ is rational normal, arithmetically normal, projectively normal, or linearly normal, using invariants like the nearly Gorenstein condition and embedding dimension at singular points.

ABSTRACT

We give refined statements and modern proofs of Rosenlicht's results about the canonical model C' of an arbitrary complete integral curve C. Notably, we prove that C and C' are birationally equivalent if and only if C is nonhyperelliptic, and that, if C is nonhyperelliptic, then C' is equal to the blowup of C with respect to the canonical sheaf ω. We also prove some new results: we determine just when C' is rational normal, arithmetically normal, projectively normal, and linearly normal.

Motivation & Objective

• To reprove and refine Rosenlicht's results on the canonical model $C'$ of a singular integral curve $C$ of genus $g \geq 2$.
• To clarify the birational relationship between $C$ and $C'$, showing they are isomorphic if and only if $C$ is nonhyperelliptic.
• To determine precise conditions under which $C'$ is projectively normal, linearly normal, arithmetically normal, or rational normal.
• To characterize the canonical model $C'$ via blowups with respect to the canonical sheaf $\omega$, especially in the non-Gorenstein case.
• To identify invariants such as the nearly Gorenstein condition and embedding dimension that control the geometric properties of $C'$.

Proposed method

• Use of modern algebraic geometry techniques, including blowups with respect to the canonical sheaf $\omega$, to construct $C'$ as the image of the canonical map on the normalization $\overline{C}$.
• Application of Castelnuovo theory and the theory of extremal curves to analyze projective normality and linear normality of $C'$.
• Employment of local algebra, particularly the conductor ideal and endomorphism rings of local rings at singular points, to study the structure of $C'$ near non-Gorenstein points.
• Use of cohomological arguments and long exact sequences in sheaf cohomology to prove vanishing results for $H^q(\mathcal{I}(l))$, which control the defining equations of $C'$.
• Proof of equivalence between projective normality, linear normality, and extremality of $C'$ under the nearly Gorenstein condition.
• Use of Fujita’s criterion and Castelnuovo–Mumford regularity to determine when $C'$ is cut out by quadrics and cubics.

Experimental results

Research questions

• RQ1 When is the canonical model $C'$ of a singular curve $C$ projectively normal, and what conditions on $C$ ensure this property?
• RQ2 Under what conditions is $C'$ linearly normal, and how does this relate to projective normality?
• RQ3 When is $C'$ rational normal, and what role does the genus and degree of $C'$ play in this?
• RQ4 How does the geometry of $C'$ depend on the singularities of $C$, particularly at non-Gorenstein points?
• RQ5 What is the precise relationship between the canonical model $C'$ and the blowup of $C$ with respect to $\omega$, and when are they isomorphic?

Key findings

• The canonical model $C'$ is isomorphic to the blowup of $C$ with respect to $\omega$ if and only if $C$ is nonhyperelliptic.
• The canonical model $C'$ is projectively normal if and only if it is linearly normal, and this holds if and only if $C$ is nearly Gorenstein and satisfies $d' = g' + g - 1$.
• When $C$ is nonhyperelliptic, $C'$ is extremal (i.e., has maximal genus for its degree $2g-2$) if and only if $C$ is Gorenstein.
• If $C$ is nearly Gorenstein and $\eta \geq 2$, then $C'$ is cut out by quadrics and cubics, with quadrics alone sufficient if $\eta \geq 2$.
• The canonical model $C'$ is rational normal if and only if $C$ is nonhyperelliptic and $C'$ lies in $\mathbb{P}^{g-1}$ with degree $g-1$ and genus $g-1$.
• For a nearly Gorenstein curve with $g=4$, $g'=2$, and $d'=5$, $C'$ cannot be cut out by quadrics alone, showing that cubics are sometimes necessary.