Let's work over $\mathbb{C}$. The degree $d$ curves in $\mathbb{P}^2_{\mathbb{C}}$ are parameterized by a projective space $|\mathcal{O}_{\mathbb{P}^2}(d)|$. Let $U_d\subset |\mathcal{O}_{\mathbb{P}^2}(d)|$ be the locus parameterizing smooth curves. Let $u\in U_{1,d}$ be a closed point, let $X_u$ be the corresponding plane curve. For any $m\in\mathbb{Z}$, the fundamental group $\pi_1(U_{1,d},u)$ acts on $\mathrm{H}^1(X_u,\mathbb{Z})$. The monodrmoy action preserves the intersection form. Let's denote the image of $\pi_1(U_{1,d},u)$ in $\mathrm{Sp}_{2g}(\mathbb{Z})$ by $\Gamma_{1,d}$.

$\bullet$When $d$ is even, by Theorem 4(i)(https://math.unice.fr/~beauvill/pubs/mono.pdf), the group $\Gamma_{1,d}=\mathrm{Sp}_{2g}(\mathbb{Z})$.

$\bullet$When $d$ is odd, Theorem 4(ii) there claimed that there exists a $\mathbb{Z}/2\mathbb{Z}$-quadratic form $q_X$ on $\mathrm{H}^1(X,\mathbb{Z}/2\mathbb{Z})\to\mathbb{Z}/2\mathbb{Z}$ such that $\Gamma_{1,d}\subset\mathrm{Sp}_{2g}(\mathbb{Z})$ is the stabilizer of $q_X$. (I think this $q_X$ comes from classification of purely algebraic structure called ``vanishing lattices'', and I am not sure what it is in explicit cases..)

$\textbf{Question}$: (1) Is this $q_X$ the one associated with the theta characteristic $\mathcal{O}_{\mathbb{P}^2}(\frac{d-3}{2})|_{X_u}$? (I think the answer is yes, as explained by abx in the comment.)

(2) If so, are there some explicit expression of generators of the group $\Gamma_{1,d}=\mathrm{Stab}_{\mathrm{Sp}_{2g}(\mathbb{Z})}(q_X)$? (Ok, I found it Reference on generators of subgroups of symplectic groups, the generators are given in Mumford Tata lectures on theta I, P208)

(3)The goal was to check $\Gamma_{1,d}$ has no common eigenvector in $\mathrm{H}^1(X,\mathbb{Z}/m\mathbb{Z})$ for every $m\in\mathbb{Z}$. It can be checked by generators displayed in (2), but I think this is a much weaker statement than determining the structure of $\Gamma_{1,d}$, is there a simpler way to check this?

[There is a quadratic form given by theta characteristic $\mathcal{N}:=\mathcal{O}_{\mathbb{P}^2}(\frac{d-3}{2})|_{X_u}$: We identify $\mathrm{H}^1(X,\mathbb{Z}/2\mathbb{Z})=\mathrm{H}^2(X,\mu_2)=\mathrm{Pic}^0(X)[2]$, then for any $2$-torsion line bundle $\mathcal{L}$, we assign the number $q(\mathcal{L})=h^0(\mathcal{L\otimes N})-h^0(\mathcal{L})$ in $\mathbb{Z}/2\mathbb{Z}$. I think $\Gamma_{1,d}$ stabilizes this $q$, but I am not sure does this mean $\Gamma_{1,d}$ is exactly the stabilizer of this $q$?]

unique$q$ preserved by the monodromy, so the form you describe is equal to $q_X$, and $\Gamma_{1,d}$ is exactly its stabilizer. $\endgroup$ – abx May 16 at 7:32