Singular curves with line bundles \(L\) defined over \(\mathbb F_q\) and with \(H^0(L)=H^1(L)=0\) (Q2853424)

Let \(Y\) be a geometrically integral projective curve defined over the finite field \(F_q\) of genus \(g\). Let \(S(Y,q)\) be the set of all degree \(p_a(Y)-1\) line bundles \(L\) on \(Y\) defined over \(F_q\) such that \(h^0(Y,L) = 0\). Suppose that \(S(Y,q) \neq \emptyset \) and \(Y_{\mathrm{reg}}(F_q) \neq \emptyset\). We fix integers \(x,y,z \geq 0\) such that \(\#(Y_{\mathrm{reg}}(F_q)) \geq 2x+y\) and \(\#(Y_{\mathrm{reg}}(F_{q^2}))\geq \#(Y_{\mathrm{reg}}(F_q))+2z\). If \(x>0\), then assume \(q \neq 2\). We denote by \(f_q\) the Frobenius map. We fix \(2x+y\) distinct points \(P_1,\dots,P_x,Q_1,\dots,Q_x,E_1,\dots,E_y\) of \(Y(F_q)\) and \(z\) distinct points \(O_1,\dots,O_z\in Y(F_{q^2})\setminus Y(F_q)\) such that \(\#(\{O_1,\dots,O_z,f_q(O_1),\dots,f_q(O_z)\}) = 2z\). Let \(X\) be the geometrically integral curve with arithmetic genus \(g+x+y+z\) and defined over \(F_q\) obtained from \(Y\) making \(y\) ordinary cusps coming from the points \(E_1,\dots,E_y\) and with \(x+z\) ordinary nodes obtained gluing together each \(P_i\) with each \(Q_i\) and each \(O_i\) with \(f_q(O_i)\). Then in this paper it is proved that the following inequality holds: NEWLINE\[NEWLINE\#(S(X,q)) \geq q^z (q-1)^y (q-2)^x \#(S(Y,q)).NEWLINE\]