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Nov 1, 2018 at 19:59 comment added user130124 The precise description of the maximal rank conjecture for curves sheds light on why this may be hard, even though the context is a little different (see arxiv.org/pdf/1711.04906.pdf for a statement of the conjecture).
Nov 1, 2018 at 19:49 comment added abx If $Y$ and $X$ are smooth and projective, you can use Hodge theory to identify your morphism to the (conjugate of) the restriction $H^0(Y,\Omega ^i_Y)\rightarrow H^0(X,\Omega ^i_X)$, which may be easier to analyze.
Nov 1, 2018 at 19:30 answer added Sasha timeline score: 7
Nov 1, 2018 at 13:43 comment added john Yes, I'm aware of that.
Nov 1, 2018 at 5:00 comment added abx Are you aware that $H^i(Y,j_*\mathcal{O}_X)$ is nothing but $H^i(X,\mathcal{O}_X)$??
Nov 1, 2018 at 0:59 history asked john CC BY-SA 4.0