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This answer of mine briefly discusses Hartshorne conjecture and some related questions about smooth subvarieties of $\mathbb P^n$ of small codimensions. It links to Hartshorne's original paper, which I think is still the best source to answer your questions 1) and 2).

As for 3), you can also look at Zolbani's thesis, which has a lot more details then his research statements mentioned by Steven.

(That's all I know, I would be very interested in what's new about Hartshorne's conjecture as well).

EDIT: Today while answering another question I was reminded of a line of research which can be viewed as evidence for Hartshorne's conjecture: smooth subvarieties of small codimension behave cohomologically like complete intersections (this was discussed in Section 2 of Hartshorne original paper). A paper by Lyubeznik, especially Section 11, has many such results, even for positive characteristic cases. It also includes many relevant references.

This answer of mine briefly discusses Hartshorne conjecture and some related questions about smooth subvarieties of $\mathbb P^n$ of small codimensions. It links to Hartshorne's original paper, which I think is still the best source to answer your questions 1) and 2).

As for 3), you can also look at Zolbani's thesis, which has a lot more details then his research statements mentioned by Steven.

(That's all I know, I would be very interested in what's new about Hartshorne's conjecture as well).

EDIT: Today while answering another question I was reminded of a line of research which can be viewed as evidence for Hartshorne's conjecture: smooth subvarieties of small codimension behave cohomologically like complete intersections (this was discussed in Section 2 of Hartshorne original paper). A paper by Lyubeznik, especially Section 11, has many such results, even for positive characteristic cases. It also includes many relevant references.

This answer of mine briefly discusses Hartshorne conjecture and some related questions about smooth subvarieties of $\mathbb P^n$ of small codimensions. It links to Hartshorne's original paper, which I think is still the best source to answer your questions 1) and 2).

As for 3), you can also look at Zolbani's thesis, which has a lot more details then his research statements mentioned by Steven.

(That's all I know, I would be very interested in what's new about Hartshorne's conjecture as well).

This answer of mine briefly discusses Hartshorne conjecture and some related questions about smooth subvarieties of $\mathbb P^n$ of small codimensions. It links to Hartshorne's original paper, which I think is still the best source to answer your questions 1) and 2).

As for 3), you can also look at Zolbani's thesis, which has a lot more details then his research statements mentioned by Steven.

(That's all I know, I would be very interested in what's new about Hartshorne's conjecture as well).

EDIT: Today while answering another question I was reminded of a line of research which can be viewed as evidence for Hartshorne's conjecture: smooth subvarieties of small codimension behave cohomologically like complete intersections (this was discussed in Section 2 of Hartshorne original paper). A paper by Lyubeznik, especially Section 11, has many such results, even for positive characteristic cases. It also includes many relevant references.

This answer of mine discuss some aspect of Hartshorne conjectures related to normality of the cone.

You can also look at Zolbani's thesis, which has a lot more details.

This answer of mine briefly discusses Hartshorne conjecture and some related questions about smooth subvarieties of $\mathbb P^n$ of small codimensions. It links to Hartshorne's original paper, which I think is still the best source to answer your questions 1) and 2).

As for 3), you can also look at Zolbani's thesis, which has a lot more details then his research statements mentioned by Steven.

(That's all I know, I would be very interested in what's new about Hartshorne's conjecture as well).