# ag.algebraic geometry – Limit of Line Bundles on Smooth Curves Degenerating to Double Line

Consider a family of smooth plane conics $$f_lambda(x,y,z)=0$$ as a family $$T_lambda = (C,L,v_1,v_2,v_3)_lambda$$ of genus zero curves with a degree 2 line bundle $$L$$ and an ordered basis $$v_i$$ for the space of global sections of $$L$$. Suppose that $$f_lambda$$ degenerates to the double line $$x^2=0$$ when $$lambda=0.$$

What are various geometric objects which could serve as a limit of $$T_lambda$$ as $$lambda$$ goes to zero, and what are some examples of situations where each limit would be (in)appropriate?

For example, the equation $$x^2=0$$ would be a possible limit that seems appropriate if we’re thinking about the $$T_lambda$$ as polynomial equations, but less appropriate if we’re thinking about the $$T_lambda$$ as $$5$$-ples $$(C,L,v_1,v_2,v_3)_lambda.$$