Let $k$ be algebraically closed and consider a nodal curve $X$ over $k$, whose irreducible components are smooth curves $X_1,\dots,X_r$. Suppose I have a line bundle on $X$ whose restriction to some components $X_1,\dots,X_a$ gives an embedding in $\mathbb{P}^n$ (i.e. it has positive degre) and that is trivial on the other components (i.e. it has zero degree). Is it true that the line bundle gives a map $X\to \mathbb{P}^n$ that is an embedding on every irreducible component of the first type and that collapses to a point those of the second type?
Maps from a nodal curve to projective space
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algebraic-geometry
algebraic-curves
riemann-surfaces