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Updated the answer in light of a better understanding of a reference
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Timothy Chow
Timothy Chow
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I suggested this candidate in a comment: For which $d$ and $g$ does there exist a curve in $\mathbb{P}^3$ of degree $d$ and genus $g$? In Hartshorne's Algebraic Geometry, Chapter VI, Section 6, it is indicated that the Steiner Prize was awarded in 1882 to Noether and Halphen for work on this problem and related problems (although if one wants to be pedantic about the earliest date on which the question was formulated explicitly, I don't know when that would be). AccordingAccording to another MO answer, Hartshorne's claim that this problem is open remains true today.

EDIT: Looking more carefully at the references in that other MO answer, I think the paper by Gruson and Peskin does actually solve this problem. (Mori is strengthening their result in certain special cases.) If so, then this problem is no longer open.

If this particular problem doesn't answer Mark Lewko's question, then there may be similar questions in classical algebraic geometry that do. For comparison, the interpolation problem, of determining when there is a (Brill–Noether) curve of degree $d$ and genus $g$ passing through $n$ general points in $\mathbb{P}^r$, was only recently solved by Larson and Vogt.

I suggested this candidate in a comment: For which $d$ and $g$ does there exist a curve in $\mathbb{P}^3$ of degree $d$ and genus $g$? In Hartshorne's Algebraic Geometry, Chapter VI, Section 6, it is indicated that the Steiner Prize was awarded in 1882 to Noether and Halphen for work on this problem and related problems (although if one wants to be pedantic about the earliest date on which the question was formulated explicitly, I don't know when that would be). According to another MO answer, Hartshorne's claim that this problem is open remains true today.

If this particular problem doesn't answer Mark Lewko's question, then there may be similar questions in classical algebraic geometry that do. For comparison, the interpolation problem, of determining when there is a (Brill–Noether) curve of degree $d$ and genus $g$ passing through $n$ general points in $\mathbb{P}^r$, was only recently solved by Larson and Vogt.

I suggested this candidate in a comment: For which $d$ and $g$ does there exist a curve in $\mathbb{P}^3$ of degree $d$ and genus $g$? In Hartshorne's Algebraic Geometry, Chapter VI, Section 6, it is indicated that the Steiner Prize was awarded in 1882 to Noether and Halphen for work on this problem and related problems (although if one wants to be pedantic about the earliest date on which the question was formulated explicitly, I don't know when that would be).According to another MO answer, Hartshorne's claim that this problem is open remains true today.

EDIT: Looking more carefully at the references in that other MO answer, I think the paper by Gruson and Peskin does actually solve this problem. (Mori is strengthening their result in certain special cases.) If so, then this problem is no longer open.

If this particular problem doesn't answer Mark Lewko's question, then there may be similar questions in classical algebraic geometry that do. For comparison, the interpolation problem, of determining when there is a (Brill–Noether) curve of degree $d$ and genus $g$ passing through $n$ general points in $\mathbb{P}^r$, was only recently solved by Larson and Vogt.

Source Link
Timothy Chow
Timothy Chow
  • 92.3k
  • 31
  • 410
  • 656

I suggested this candidate in a comment: For which $d$ and $g$ does there exist a curve in $\mathbb{P}^3$ of degree $d$ and genus $g$? In Hartshorne's Algebraic Geometry, Chapter VI, Section 6, it is indicated that the Steiner Prize was awarded in 1882 to Noether and Halphen for work on this problem and related problems (although if one wants to be pedantic about the earliest date on which the question was formulated explicitly, I don't know when that would be). According to another MO answer, Hartshorne's claim that this problem is open remains true today.

If this particular problem doesn't answer Mark Lewko's question, then there may be similar questions in classical algebraic geometry that do. For comparison, the interpolation problem, of determining when there is a (Brill–Noether) curve of degree $d$ and genus $g$ passing through $n$ general points in $\mathbb{P}^r$, was only recently solved by Larson and Vogt.