Let $C \rightarrow \mathbb P^6$ be a genus 7 canonically embedded (singular) Gorenstein generically reduced curve. Are there any examples of such a curve so that the conormal sheaf $N^\vee\_{C/\mathbb P^6} \cong \mathscr I\_C/ \mathscr I\_C^2$ is not locally free, or do all genus 7 Gorenstein canonical curves have locally free conormal sheaves? 


For example, if C is a regular embedding, then it is not hard to show the conormal sheaf is locally free. See, for instance, Qing Liu's Algebraic Geometry and Arithmetic Curves, Section 6.3, Corollary 3.8. So, any regular embedding $C \rightarrow \mathbb P^6$ would yield a locally free conormal sheaf.


If an example isn't known in genus 7, I'd also be interested in seeing an example in another genus.



Example of Genus 7 Curve whose Conormal Sheaf isn't Locally Free
There are examples already beginning in genus $5$, and thus in all higher genera (including genus 7). By Serre's Corollary, Corollary 21.20 of the following, you **cannot** find examples with genus $3$ or $4$ (the problem does not make sense for $g=2$, since the canonical map cannot be a closed immersion). The construction below is based on Exercise 21.11, p. 553, of Eisenbud's book.


MR1322960 (97a:13001)   

Eisenbud, David(1-BRND)   

Commutative algebra. With a view toward algebraic geometry.   

Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp.   
 
ISBN: 0-387-94268-8; 0-387-94269-6   

13-01 (14A05)   



There exists a finite morphism of schemes $f:\mathbb{P}^1\to C$ that is a universal homeomorphism, that is an isomorphism on the dense open subset $\mathbb{P}^1\setminus \{0\}$, and such that the induced finite ring homomorphism,
$$ \widehat{f}^{\#}\_0: \widehat{\mathcal{O}}\_{C,0}\to \widehat{\mathcal{O}}\_{\mathbb{P}^1,0} $$
is equivalent to the finite ring homomorphism
$$
k[[t^5,t^6,t^7,t^8]]\hookrightarrow k[[t]],
$$
where $t$ is a generator of the maximal ideal $\mathfrak{m}\_{\mathbb{P}^1,0} \subset \widehat{\mathcal{O}}\_{\mathbb{P}^1,0}$. The scheme $C$ is Gorenstein by Exercise 21.11. 


The embedding dimension of $C$ at the origin is $4$ with one minimal set of generators of $\mathfrak{m}\_{C,0}$ given by $$x\_5 = t^5,\ x\_6 = t^6,\ x\_7 = t^7,\ x\_8 = t^8.$$ For the induced surjection $$ \phi^\*: k[[x\_5,x\_6,x\_7,x\_8]] \to k[[t^5,t^6,t^7,t^8]],\ \ x\_n \mapsto t^n,$$ for the associated ideal $I=\text{Ker}(\phi)$, $I/\mathfrak{m}I$ contains the following $k$-linearly independent elements, $$ g\_1 = x\_5x\_7-x\_6^2,\ g\_2 = x\_5x\_8-x\_6x\_7, \ g\_3 = x\_6x\_8-x\_7^2, \ g\_4 = x\_7x\_8 - x\_5^3,\ g\_5 = x\_8^2-x\_5^2x\_6.$$ (This might be a $k$-basis; as it is irrelevant to what follows, I did not compute this.) Thus $C$ is not a local complete intersection scheme. 


Now choose $t$ to be an element of $k(\mathbb{P}^1)$ that is regular on $\mathbb{P}^1\setminus\{\infty\}$, that has a simple pole at $\infty$ and that has a simple zero at $0$, i.e., $t$ is a usual coordinate on $\mathbb{A}^1 = \mathbb{P}^1\setminus\{\infty\}$. Let $\alpha$ be the rational section $dt/t^2$ of $\omega\_{\mathbb{P}^1/k}=\Omega^1\_{\mathbb{P}^1/k}$. Thus $\alpha$ is a regular generator of $\omega\_{\mathbb{P}^1/k}$ on the open subset $\mathbb{P}^1\setminus\{0\}$, and $\alpha$ has a pole of order $2$ at $0$. By Exercise 21.11, $H^0(C,\omega\_{C/k})$ is generated by $$ \beta\_0 = \frac{1}{t^8}\alpha, \ \beta\_1 = x\_5\beta\_0 = \frac{1}{t^3}\alpha, \ \beta\_2 = x\_6\beta\_0 = \frac{1}{t^2}\alpha,\ \beta\_3 = x\_7\beta\_0 = \frac{1}{t}\alpha,\ \beta\_4 = x\_8\beta\_0 = \alpha.$$


Moreover, $\beta\_0$ is a generator of $\omega\_{C/k}$ on $C\setminus\{\infty\}$, and $\beta\_4$ is a generator of $\omega\_{C/k}$ on $C\setminus\{0\}$. Thus the canonical morphism $$\phi:C \to \mathbb{P}H^0(C,\omega\_{C/k}),$$ restricts on the open affine $C\setminus\{\infty\} \to D\_+(\beta\_0)$ to the morphism of schemes associated to the ring homomorphism $\phi^\*$ above. So $\mathcal{I}\_C/\mathcal{I}\_C^2$ is not locally free of rank $3$: the rank at $0$ is (at least) $5$.


This example has arithmetic genus $5$. You can make an example with arithmetic genus $7$ in various ways, e.g., glue a copy of a curve of arithmetic genus $2$ at a point $p\neq 0$ of $C$ such that the total curve has an ordinary double point at $p$. 


P.S. For the one or two people who know what I am talking about, this example solves the extra credit problem from Hartshorne's 1995 commutative algebra final -- not quite completed by the exam deadline :)


