Let $X$ be a $K3$ surface, $\rm{Pic}(X)=\mathbb{Z}\cdot H$ where $H^2=2$, $C\in |2H|$ be smooth, $L$ a degree $6$ line bundle on $C$ and $i:C\to X$ the inclusion.

I wanted to compute the first and second Chern classes of the torsion sheaf $i_*L$ on $X$.

I was once told that everything can be computed using Grothendieck-Riemann-Roch, especially when there is a closed embedding.

Perfect, I've used it and I've found $c_1(i_*L)=2H$ and $c_2(i_*L)=2$.

Anyway, it seems to me that I have overkilled the problem and that in fact I have lost all the geometry behind this computation. I am pretty sure there must exist a easier (= more geometric) solution to arrive to the same result without using G-R-R.

I'd appreciate any comment about different approaches.

Thank you very much!

P.S.: I have studied G-R-R on Fulton and the technique he uses is the so called "deformation of the normal bundle". I think my lack of a geometric meaning of the computation above sits inside my non-understanding of this technique (that I still look as a magical technical detail..). I hope some of the proposed alternative computations can help me with this bigger gap.


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