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1. Initial Problem Session

    1.     The goal is to formulate a Manin-type conjecture for degree $d$ points on Fano varieties. Let $X/k$ be a Fano variety defined over a number field $k$. For a subset $U\subset X$, an integer $d$, a real number $B>0$, and a line bundle $\mathscr L$ on $X$, define $$N_{X,\mathscr L}(U,B;d) := \#\left\{x\in U(\overline k):h_{\mathscr L}(x)\le B\text{ and }[\kappa(x):k]=d\right\},$$ where $h_{\mathscr L}$ is a Weil height associated to $\mathscr L$. For appropriately chosen $U$, one may expect that $N_{X,\mathscr L}(U,B;d)\sim cB^a(\log B)^b$ as $B\to\infty$, for some choice of $a,b,c$.

      Add remark

      Problem 1.02.

      [Caleb Ji] There were several questions proposed around this, including

      * What should the predicted values of $a,b,c$ be?

      * What is the "shape" of the set $U$ that one should take? In the usual Manin conjecture, one expects that $U$ can be taken to be the complement of a thin set.

      * Can one compute asymptotics for $N_{X,\mathscr L}(U,B;d)$ if $X=\mathbb P^2_{\mathbb Q}$ and $d=3$, for some choices of $\mathscr L$?
        1. Remark. The case of $X=\mathbb P^2_{\mathbb Q}$ and $d=2$ has been studied in an old paper of Schmidt. One can also see Cécile Le Rudulier’s thesis (or the related paper [MR4057715]) which also handles the case of $\mathbb P^1_{\mathbb Q}\times\mathbb P^1_{\mathbb Q}$ and $d=2$. There is also a recent paper of Jesse Kass and Frank Thorne [arXiv:2209.13030] which counts points on $\operatorname{Hilb}^2(\mathbb P^2_{\mathbb Q})$.

          Finally, a similar problem (for $d=2$) was studied last summer as part of an IAS summer collaboration; find the report at https://www.ias.edu/sites/default/files/Park%20report.pdf


            • Add remark

              Problem 1.04.

              [Adam Logan] Is there a $\mathbb Q$-irrational del Pezzo surface $X$ which is $\mathbb Q$-unirational for which $\operatorname{Sym}^dX$ is $\mathbb Q$-rational for some $d>1$?
                1. Remark. [Adam Logan] $X$ should have negative Kodaira dimension for this to be possible.
                    • Remark. [Nathan Chen] If $X$ is $k$-rational, then so is $\operatorname{Sym}^d(X)$. Over $\mathbb C$, Arapura showed that the Kodaira dimension of $\operatorname{Sym}^d(X)$ is $$\kappa(\operatorname{Sym}^d(X))=d\kappa(X)$$ if $\dim X\ge2$.
                        • Remark. [Bianca Viray] While this example is not unirational, if $X=C\times\mathbb P^1$ with $C$ a conic, then $X$ is irrational while $\operatorname{Sym}^2(X)$ is rational.
                            • Remark. [Bianca Viray] As a followup, can we characterize the $X$ (and $d$) for which the answer to Problem 1.2 is yes?
                                • Remark. [Lena Ji] If $X$ is a non-trivial Severi-Brauer variety of index $d$, then $X$ is not rational, but $\operatorname{Sym}^d(X)$ is rational.
                                    • Remark. [Morena Porzio] One could try looking at a cubic surface $X$ with $d=3$ to see if this gives an example.

                                        •     Let $X$ be a variety. Suppose one has a diagram $X\dashleftarrow Y\dashrightarrow\mathbb P^n$, where $Y\dashrightarrow\mathbb P^n$ is finite of degree $d$. Then, as a consequence of Hilbert irreducibility, $X$ will have a Zariski dense set of degree $d$ points.

                                          Add remark

                                          Problem 1.06.

                                          [Isabel Vogt + Olivier Martin] Can we give a method to construct a Zariski dense set of degree $d$ points on a variety different from the one just mentioned?

                                          * Maybe one can replace $\mathbb P^n$ with another sort of variety?

                                          * Maybe one can do something totally different and give a construction not coming from a correspondence.
                                            1. Remark. [Bianca Viray] It would be nice to write down a census of all the ways we know to produce a Zariski dense set of degree $d$ points on varieties. This leads to the subquestions:

                                              * Is the method of using AV parameterized points (on curves) genuinely different than a method using correspondences?

                                              * Is there a construction that does not come from curves (or correspondences)?
                                                • Remark. [Borys Kadets] In a sense, everything comes from a correspondence between $X$ and $\operatorname{Sym}^dX$, so you need to be careful about which correspondences you consider.
                                                    • Remark. [Adam Logan] Suggested other construction. Take a K3 surface $X$ with infinite automorphism group, but which contains no smooth curves of genus $0$ or $1$. Given a degree $d$ point on $X$, its orbit under the automorphism group will be infinite and likely will be Zariski dense in $X$ (since it cannot lie on a smooth curve in $X$). To really make this work, one might need the stronger-seeming condition that $X$ contains no (possibly singular) curves of geometric genus $0$ or $1$ defined over the ground field. This likely holds for examples one can construct satisfying the initial condition of no smooth low genus curves, but it may be harder to provably verify this.


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                                                          Problem 1.08.

                                                          [Morena Porzio + Isabel Vogt] With some hypotheses on $X$ and $X\to\operatorname{Alb}(X)$, study the image of $\operatorname{Sym}^d(X)\to\operatorname{Alb}(X)$ for some small $d$.

                                                          What are examples of $X$ for which the fibers of $\operatorname{Sym}^dX\to\operatorname{Alb}(X)$ are "well-understood"? For example, when are they finite or rational?
                                                            1. Remark. [Isabel Vogt] For such $X$, one could hope to apply Mordell-Lang in order to better understand the structure of degree $d$ points, in analogy with the case of points on curves.
                                                                • Remark. [Olivier Martin] Maybe consider $F_1(X)$, the Fano variety of lines on a cubic $3$-fold.
                                                                    • Remark. [Jesse Wolfson] Maybe restrict to $X$ whose image in $\operatorname{Alb}(X)$ is of maximal dimension.
                                                                        • Remark. [Morena Porzio] Maybe consider $X$ for which $\operatorname{Alb}(X)$ is simple.

                                                                            •     Let $X/k$ be a nice variety. We define a few measures of irrationality.

                                                                              * Its covering gonality $\operatorname{cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\leftarrow\mathcal C\dashrightarrow X$, where $\mathcal C\dashrightarrow X$ is a dominant rational map and $\mathcal C\to B$ is a family of curves such that $\operatorname{gon}(\mathcal C_b)\ge d$ for all $b\in B$.

                                                                              * Its connected covering gonality $\operatorname{conn.cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\leftarrow\mathcal C\dashrightarrow X$, where $\mathcal C\to B$ is a family of curves with $\operatorname{gon}(\mathcal C_b)\ge d$ for all $b\in B$ and such that the induced rational map $\mathcal C\times_B\mathcal C\dashrightarrow X\times X$ is dominant.

                                                                              * Finally, $\operatorname{uni.irr}(X)$ is the minimal $d$ such that there exists a diagram $X\dashleftarrow Y\dashrightarrow\mathbb P^n$ with $Y\dashrightarrow X$ dominant and $Y\dashrightarrow\mathbb P^n$ finite of degree $d$.

                                                                              Add remark

                                                                              Problem 1.1.

                                                                              [Olivier Martin] Let $X/k$ be a nice variety, and let $\delta(X/k)=\{d:$ degree $d$ points on $X$ are Zariski dense in $X\}$ be its density degree set. How different can $\min\delta(X/k)$ be from $\operatorname{uni.irr}(X),\operatorname{cov.gon}(X)$, and/or $\operatorname{conn.cov.gon}(X)$?
                                                                                1. Remark. [Borys Kadets] Consider $X=Y\times\mathbb P^1$ with $Y$ a complicated variety. Then, $\operatorname{cov.gon}(X)=1$, but $\min\delta(X/k)=\min\delta(Y/k)$, so it seems that $\min\delta(X/k)$ and $\operatorname{cov.gon}(X)$ can differ by quite a lot in general.

                                                                                    •     Let $X/k$ be a nice variety. In addition to its density degree set $\delta(X/\mathbb Q)$, define $$\mathcal D(X/\mathbb Q):=\{d:X\text{ has a point of degree }d\},$$ and $$\mathcal P(X/\mathbb Q)=\bigcup_{[k':k]<\infty}\delta(X/k').$$

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                                                                                      Problem 1.12.

                                                                                      [Valia Gazaki + Bianca Viray] Skorobogatov constructed a bielliptic surface $X/\mathbb Q$ such that $X(\mathbb A_{\mathbb Q})^{\operatorname{Br}}\neq\emptyset$, but $X(\mathbb Q)=\emptyset$. It is known that $\min\delta(X/\mathbb Q)\le3$. Where do these Zariski-dense degree 3 points come from? Can one compute $\mathcal D(X/\mathbb Q),\delta(X/\mathbb Q),\mathcal P(X/\mathbb Q)$?


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                                                                                          Problem 1.14.

                                                                                          [Nathan Chen + Olivier Martin] Let $C,D$ be hyperelliptic curves of genus $\ge2$ over a number field $k$. Must $2\in\delta(C\times D/k)$? Must it be in $\mathcal P(C\times D/k)$?

                                                                                          More generally, let $Z\subset C\times D$ be the closure of the degree $d$ points. When is there a component of $Z$ which dominates both $C$ and $D$?


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                                                                                              Problem 1.16.

                                                                                              [James Rawson] Let $C$ be a curve of genus $\ge2$ over a number field. Must $2\in\delta(\operatorname{Sym}^2_C/k)$ or in $\mathcal P(\operatorname{Sym}^2_C/k)$? What do these sets look like?


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                                                                                                  Problem 1.18.

                                                                                                  [Adam Logan + Bianca Viray] Given a finite map $f:X\to Y$ (over a number field), how are $\delta(X/k)$ and $\delta(Y/k)$ related? Are there examples showing that the obvious relations are sharp?
                                                                                                    1. Remark. [Nathan Chen] It could be interesting to consider the special case where $f$ is a Galois cover.
                                                                                                        • Remark. [Morena Porzio] If $X,Y$ are curves, you can also ask about containment relations between there $\delta_{\mathbb P^1}$’s and their $\delta_{\operatorname{AV}}$’s.


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                                                                                                              Problem 1.2.

                                                                                                              [Morena Porzio + Isabel Vogt] Is $\operatorname{Hilb}^2(C\times D)(k)$ Zariski dense in $\operatorname{Hilb}^2(C\times D)$, where $C,D/k$ are hyperelliptic curves over a number field. More generally, for which varieties $X$ over a number field is there a $d$ such that $\operatorname{Hilb}^d(X)(k)$ is Zariski dense?
                                                                                                                1. Remark. [Nathan Chen] If $X$ is of general type and $\dim X\ge2$, then $\operatorname{Hilb}^d(X)$ will also be of general type.
                                                                                                                    • Remark. [Niven Achenjang + Caleb Ji] One can also ask, is there a $d$ such that the set $\{$degree $d$ points of $X\}\subsetneq\operatorname{Hilb}^d(X)$ is Zariski dense in $\operatorname{Hilb}^d(X)$


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                                                                                                                          Problem 1.22.

                                                                                                                          [Shamil Asgarli + Bianca Viray] Given some specific geometric construction showing that $d\in\delta(X/k)$, can one use it to construct a dominant rational map $X\dashrightarrow\mathbb P^2$ of degree $nd$, for some small $n$?


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                                                                                                                              Problem 1.24.

                                                                                                                              [Nathan Chen] Given a covering family $T\leftarrow\mathcal C\dashrightarrow X$ of curves, where $T(k)$ is Zariski dense, how does $\operatorname{gon}(C)$ (for $C$ a generic member of the family) compare to $\min\delta(X/k)$?

                                                                                                                                •     For a variety $X/F$, define its degree of irrationality $\operatorname{irr}(X)$ to be the minimum degree of a rational map $X\dashrightarrow\mathbb P^2$ (defined over $F$).

                                                                                                                                  Add remark

                                                                                                                                  Problem 1.26.

                                                                                                                                  [Nathan Chen] Compute $\operatorname{irr}(X/F)$ for a del Pezzo surface $X$ over a general field $F$.


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                                                                                                                                      Problem 1.28.

                                                                                                                                      [Nathan Chen + Lena Ji] How do all of these invariants (e.g. $\min\delta(X/k),\operatorname{irr}(X/k),\operatorname{cov.gon}(X/k)$, etc.) behave under specialization/deformation in a smooth family? This includes the arithmetic case, where the base is something like $\operatorname{Spec}\mathscr O_{k,S}$.
                                                                                                                                        1. Remark. [Nathan Chen] Over $k=\overline k$, it is known that covering gonality goes down in specializations. This is known for degree of irrationality in smooth families of simply connected surfaces (over $\mathbb C$).
                                                                                                                                            • Remark. [Isabel Vogt] $\delta(-)$ is not very interested over finite fields because there are always only finitely many degree $d$ points.


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                                                                                                                                                  Problem 1.3.

                                                                                                                                                  [Isabel Vogt] Is there a surface $X$ over a number field $k$ and a finite extension $k'/k$ such that $\delta(X/k)\not\subset\delta(X/k')$?
                                                                                                                                                    1. Remark. [Isabel Vogt] If $X$ is a curve, then this can not happen.

                                                                                                                                                        •     The following is asked as an analogue of a result due to Debarre and Klassen [arXiv:alg-geom/9210004].

                                                                                                                                                          Add remark

                                                                                                                                                          Problem 1.32.

                                                                                                                                                          [Olivier Martin] Let $X_d\subset\mathbb P^3_k$ ($k$ a number field) be a smooth surface of degree $d$. Are the degree $\le d-3$ points not Zariski dense? What is $\min\delta(X_d/k)$?
                                                                                                                                                            1. Remark. [Bianca Viray] You probably want to assume that $d$ is sufficiently large.
                                                                                                                                                                • Remark. [Borys Kadets] One can also ask this question for the generic degree $d$ hypersurface, defined over the field of rational functions in the appropriate number of variables.
                                                                                                                                                                    • Remark. [Bianca Viray] For $d$ small, can we construct surfaces such that the degree $\le d-3$ points *are* Zariski dense?


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                                                                                                                                                                          Problem 1.34.

                                                                                                                                                                          [Olivier Martin] Given a dominant map $\psi:X\dashrightarrow\mathbb P^{\dim X}$ of minimal degree $d$, is the gonality of the pullback of a general line equal to $d$?


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                                                                                                                                                                              Problem 1.36.

                                                                                                                                                                              [Shamil Asgarli] Given a general smooth degree $d$ hypersurface $X_d\subset\mathbb P^3_k$, for $d$ sufficiently large, must a general degree $d$ point arise from intersecting $X$ with a line?

                                                                                                                                                                                •     The following is asked in analogy with a result of Harris and Silverman showing that, for a curve $C/k$, $2\in\delta(C/k)\iff C$ is a double cover of $\mathbb P^1$ or a positive rank elliptic curve (see https://www.ams.org/journals/proc/1991-112-02/S0002-9939-1991-1055774-0/S0002-9939-1991-1055774-0.pdf).

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                                                                                                                                                                                  Problem 1.38.

                                                                                                                                                                                  [Nathan Chen] Classify surfaces $X/k$ of general type with $2\in\delta(X/k)$, possibly using geometric and/or dynamical constructions.
                                                                                                                                                                                    1. Remark. [Isabel Vogt] Vojta has prove that for a hyperelliptic curve $C/k$ of genus $\ge4$, all but finitely many degree $2$ points are contracted by its hyperelliptic map.
                                                                                                                                                                                        • Remark. [Isabel Vogt] If $\operatorname{irr}(X)=2$, there exists a dominant map $X\dashrightarrow\mathbb P^2$ of degree $2$. What conditions on $X$ guarantee that the set of quadratic points on $X$ not arising from this map is not Zariski dense?


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                                                                                                                                                                                              Problem 1.4.

                                                                                                                                                                                              [Jesse Wolfson] 1.) Find effective obstructions to the existence of solvable points on varieties (in any characteristic).

                                                                                                                                                                                              2.) Do there exist curves (over surfaces) over $\mathbb Q$ without solvable points?
                                                                                                                                                                                                1. Remark. [Jesse Wolfson] There are counterexamples for curves in positive characteristic for large enough genus. See, for example, Pál’s paper https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/solvable-points-on-projective-algebraic-curves/ECE9AF2C2ED1C85F7733930C0544361C
                                                                                                                                                                                                    • Remark. [Borys Kadets] One could ask similar questions with ’solvable’ replaced by ’cyclic’ or ’abelian’.

                                                                                                                                                                                                          Cite this as: AimPL: Degree d points on algebraic surfaces, available at http://aimpl.org/degreedsurface.