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awarded  Enlightened 
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awarded  Nice Answer 
Mar 16 
comment 
Mixed Hodge structure and cup product
I had similar thoughts. I wonder how many people who voted to close knew the answer. (Test: Is the category of mixed Hodge structures Tannakian, true of false?) 
Mar 14 
comment 
Reference Request: Fundamental Group Scheme
I agree that the question seems reasonable and doesn't deserve to be closed. However, my suggestion to Priyankur would be to learn a bit more algebraic geometry before jumping into this topic. At the very least, first learn a bit about the etale fundamental group from Murre's "Lectures on Grothendieck's fundamental group" TIFR. 
Mar 9 
comment 
Terminology regarding divisor on a curve
The "support of $D$" sounds good to me. 
Feb 26 
comment 
Is there an analogue of the Tate (and Hodge) conjecture for varieties that are not proper smooth (i.e., the mixed case)?
If $X$ is smooth at least, you can use duality to flip to a cohomological statement. Then it would a special case. This is actually what I meant by my somewhat cryptic comment "expected to hold for $i=2p$ (Hodge/Tate)" 
Feb 25 
comment 
Is there an analogue of the Tate (and Hodge) conjecture for varieties that are not proper smooth (i.e., the mixed case)?
Yes. A triangulated category is all you need for the $Ext$'s. 
Feb 25 
revised 
Is there an analogue of the Tate (and Hodge) conjecture for varieties that are not proper smooth (i.e., the mixed case)?
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Feb 25 
comment 
Is there an analogue of the Tate (and Hodge) conjecture for varieties that are not proper smooth (i.e., the mixed case)?
OK, I'll try to expand this a bit later when I have more time. 
Feb 25 
answered  Is there an analogue of the Tate (and Hodge) conjecture for varieties that are not proper smooth (i.e., the mixed case)? 
Feb 23 
answered  Proof of “generic curve of genus at least 2 has no nontrivial maps to a positive genus curve” 
Feb 23 
comment 
Proof of “generic curve of genus at least 2 has no nontrivial maps to a positive genus curve”
For your first question, it follows from from Cor. 17.5.2 of Birkenhake and Lange's book on Complex Abelian Varieties (2nd ed), but I'm sure there are more suitable references. 
Feb 23 
comment 
Proof of “generic curve of genus at least 2 has no nontrivial maps to a positive genus curve”
Of course, Jason is correct. However, a very general point of $M_g$ (by which I mean, exclude a countable union of proper sub varieties) corresponds to a curve with a simple Jacobian. This curve can't map onto anything of smaller genus. (This is over $\mathbb{C}$.) 
Feb 21 
awarded  Yearling 
Feb 18 
answered  Hodge structure of relative cohomology groups 
Jan 29 
comment 
de Rham cohomology of smooth affine varieties
Not really, but this is not a bad thing. I added some further comments. 
Jan 29 
revised 
de Rham cohomology of smooth affine varieties
added 578 characters in body 
Jan 29 
answered  de Rham cohomology of smooth affine varieties 
Jan 19 
answered  periods of Mixed Hodge Structures 
Jan 13 
awarded  Nice Answer 