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Actually, a very similar statement can be found in the paper Reflexive pull-backs and base extension by Brendan Hassett and myself. See Proposition 3.5. Indeed you do not need normality, only that the fibers are $S_2$ and the sheaf does not need to be a line bundle only coherent and flat over the base. We develop a little bit of a relative theory in section 3, so you might find the rest useful as well, especially the statement regarding a characterization via local cohomology. We did assume the codimension two condition for every fiber, but it might not be necessary actually. I would have to check the proof (I will try to do that sometime). I would add that (as you discovered) for a Hartogs type statement you do not need normality and many things can be done for $S_2$ schemes that are usually done for normal ones. The main reason normality is more widely used (besides that it is easier to define and think that one understands it better) is that working with divisors on $S_2$ but not normal schemes has to be done very carefully. See this MO answer for more on this and perhaps this and this for more on Hartogs type questions.

Actually, a very similar statement can be found in the paper Reflexive pull-backs and base extension by Brendan Hassett and myself. See Proposition 3.5. Indeed you do not need normality, only that the fibers are $S_2$ and the sheaf does not need to be a line bundle only coherent and flat over the base. We develop a little bit of a relative theory in section 3, so you might find the rest useful as well, especially the statement regarding a characterization via local cohomology. We did assume the codimension two condition for every fiber, but it might not be necessary actually. I would have to check the proof (I will try to do that sometime). I would add that (as you discovered) for a Hartogs type statement you do not need normality and many things can be done for $S_2$ schemes that are usually done for normal ones. The main reason normality is more widely used (besides that it is easier to define and think that one understands it better) is that working with divisors on $S_2$ but not normal schemes has to be done very carefully. See this MO answer for more on this and perhaps this and this for more on Hartogs type questions.

Actually, a very similar statement can be found in the paper Reflexive pull-backs and base extension by Brendan Hassett and myself. See Proposition 3.5. Indeed you do not need normality, only that the fibers are $S_2$ and the sheaf does not need to be a line bundle only coherent and flat over the base. We develop a little bit of a relative theory in section 3, so you might find the rest useful as well, especially the statement regarding a characterization via local cohomology. We did assume the codimension two condition for every fiber, but it might not be necessary actually. I would have to check the proof (I will try to do that sometimes). I would add that (as you discovered) for a Hartogs type statement you do not need normality and many things can be done for $S_2$ schemes that are usually done for normal ones. The main reason normality is more widely used (besides that it is easier to define and think that one understands it better) is that working with divisors on $S_2$ but not normal schemes has to be done very carefully. See this MO answer for more on this and perhaps this and this for more on Hartogs type questions.

Actually, a very similar statement can be found in the paper Reflexive pull-backs and base extension by Brendan Hassett and myself. See Proposition 3.5. Indeed you do not need normality, only that the fibers are $S_2$ and the sheaf does not need to be a line bundle only coherent and flat over the base. We develop a little bit of a relative theory in section 3, so you might find the rest useful as well, especially the statement regarding a characterization via local cohomology. We did assume the codimension two condition for every fiber, but it might not be necessary actually. I would have to check the proof (I will try to do that sometime). I would add that (as you discovered) for a Hartogs type statement you do not need normality and many things can be done for $S_2$ schemes that are usually done for normal ones. The main reason normality is more widely used (besides that it is easier to define and think that one understands it better) is that working with divisors on $S_2$ but not normal schemes has to be done very carefully. See this MO answer for more on this and perhaps this and this for more on Hartogs type questions.

Actually, a very similar statement can be found in the paper Reflexive pull-backs and base extension by Brendan Hassett and myself. See Proposition 3.5. Indeed you do not need normality, only that the fibers are $S_2$ and the sheaf does not need to be a line bundle only coherent and flat over the base. We develop a little bit of a relative theory in section 3, so you might find the rest useful as well, especially the statement regarding a characterization via local cohomology. We did assume the codimension two condition for every fiber, but it might not be necessary actually. I would have to check the proof (I will try to do that sometimes). I would add that (as you discovered) for Hartogs type statement you do not need normality and many things can be done for $S_2$ schemes that are usually done for normal ones. The main reason normality is more widely used (besides that it is easier to define and think that one understands it better) is that working with divisors on $S_2$ but not normal schemes has to be done very carefully. See this MO answer for more on this and perhaps this and this for more on Hartogs type questions.

Actually, a very similar statement can be found in the paper Reflexive pull-backs and base extension by Brendan Hassett and myself. See Proposition 3.5. Indeed you do not need normality, only that the fibers are $S_2$ and the sheaf does not need to be a line bundle only coherent and flat over the base. We develop a little bit of a relative theory in section 3, so you might find the rest useful as well, especially the statement regarding a characterization via local cohomology. We did assume the codimension two condition for every fiber, but it might not be necessary actually. I would have to check the proof (I will try to do that sometimes). I would add that (as you discovered) for a Hartogs type statement you do not need normality and many things can be done for $S_2$ schemes that are usually done for normal ones. The main reason normality is more widely used (besides that it is easier to define and think that one understands it better) is that working with divisors on $S_2$ but not normal schemes has to be done very carefully. See this MO answer for more on this and perhaps this and this for more on Hartogs type questions.