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Suppose that $(X,x)$ be pointed nonsingular curve and we have two families over $X$ of stable maps to$P^r$: $(\pi:C\longrightarrow X,{p_i},f)$ ($(\pi^{'}:C\longrightarrow X,{p_i}^{'},f^{'})$ If we …

Suppose that $K=D(A;B;D_1;D_2)$ be a boundry divisor of $\overline{M}_{0,n}(X,\beta)$ and let $\overline{M}_A=$ $\overline{M}_{0,A \cup \bullet}(X,\beta)$ where $\bullet$ is additional marking.let $e_ …

I read in mirror symmetry and algebraic geometry by Cox and Katz that we have stable map $f : C \to Y$ which C is nodal curve with $n$ marked point then we have $0 \to Ext^0_C([f^*\Omega_Y \to \Omega …

Suppose that $C\cong P^1$ and $Def(f)$ denote the first order deformation of pointed stable map $(C,{p_i},f:C\longrightarrow X)$. I read that we have short exact sequence: $0\longrightarrow H^0(C,T_C …

In geometry of algebraic curve by Arbarello,Cornalba and Griffiths they difine trace map of dualizing sheaf of nodal curve as follows: we choose $D=r_1 +r_2+...+r_h$ consisting of h distinct smooth …