# Fixed-domain curve counts for blow-ups of projective space

Alessio Cela, Carl Lian

###### Abstract

We study the problem of counting pointed curves of fixed complex structure in
blow-ups of projective space at general points. The geometric and virtual
(Gromov-Witten) counts are found to agree asymptotically in the Fano (and some
$(-K)$-nef) examples, but not in general. For toric blow-ups, geometric counts
are expressed in terms of integrals on products of Jacobians and symmetric
products of the domain curves, and evaluated explicitly in genus 0 and in the
case of ${\mathsf{Bl}}_{q}(\mathbb{P}^{r})$. Virtual counts for
${\mathsf{Bl}}_{q}(\mathbb{P}^{r})$ are also computed via the quantum
cohomology ring.

###### Contents

  1. 1 Introduction
    1. 1.1 Curve-counting with fixed domain
    2. 1.2 Curve counts on $\mathbb{P}^{r}$
    3. 1.3 New results
      1. 1.3.1 Enumerativity
      2. 1.3.2 Geometric calculations
      3. 1.3.3 Virtual calculations
    4. 1.4 Acknowledgments
  2. 2 Enumerativity
    1. 2.1 Failure of SAE
    2. 2.2 Generalities for Fano varieties
    3. 2.3 Blow-ups of $\mathbb{P}^{r}$
    4. 2.4 SAE for ${\mathsf{Bl}}_{q}(\mathbb{P}^{r})$
    5. 2.5 SAE for del Pezzo surfaces
    6. 2.6 SAE in dimension $3$
      1. 2.6.1 Low degree effective curves in $\mathbb{P}^{3}$
      2. 2.6.2 Proof of SAE
  3. 3 Geometric counts
    1. 3.1 Integral formula on $\mathbb{P}$
      1. 3.1.1 Interlude on the permutohedral variety
      2. 3.1.2 Transversality
    2. 3.2 Integral formula on $S$
      1. 3.2.1 Cohomology of $\mathsf{Jac^{d}}(C)\times\mathsf{Sym}^{k_{1}}(C)\times\cdots\times\mathsf{Sym}^{k_{r+1}}(C)$
      2. 3.2.2 Computation of $c(\widetilde{\mathcal{E}})$
    3. 3.3 Specialization to genus 0
    4. 3.4 Specialization to $X={\mathsf{Bl}}_{q}(\mathbb{P}^{r})$
  4. 4 Virtual counts for $X={\mathsf{Bl}}_{q}(\mathbb{P}^{r})$
    1. 4.1 Preliminaries on $QH^{*}(X)$
    2. 4.2 The quantum Euler class of $X$
    3. 4.3 Proof of Theorem 14

##  1 Introduction

###  1.1 Curve-counting with fixed domain

Let $X$ be a nonsingular, irreducible, projective variety over $\mathbb{C}$ of
dimension $r$. Curve-counting problems on $X$ are traditionally formulated in
Gromov-Witten theory as intersection numbers on the space of stable maps
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ against the virtual fundamental class

$[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}\in
A_{\mathrm{vdim}(\overline{\mathcal{M}}_{g,n}(X,\beta))}(\overline{\mathcal{M}}_{g,n}(X,\beta)).$

For example, fixing $n$ subvarieties $X_{i}\subset X$, pulling back the
classes of the $X_{i}$ under the evaluation maps
$\mathrm{ev}_{i}:\overline{\mathcal{M}}_{g,n}(X,\beta)\to X$ and integrating
the product against the virtual class gives a virtual count of the number of
genus $g$ curves of class $\beta$ passing through the $X_{i}$.

We will be concerned with such problems where, in addition, the complex
structure of the domain curve $(C,p_{1},\ldots,p_{n})$ is fixed (and general).
Gromov-Witten counts are therefore obtained by additionally restricting to the
pullback of a point under the forgetful map
$\overline{\mathcal{M}}_{g,n}(X,\beta)\to\overline{\mathcal{M}}_{g,n}$.

There are two salient features of the fixed-domain version of the problem:
first, degenerating $(C,p_{1},\ldots,p_{n})$ to a union of rational curves,
the answers may be expressed purely in terms of the small quantum cohomology
ring of $X$ [4, Theorem 1.3], and are therefore tend to be simpler than
arbitrary Gromov-Witten invariants. Second, the restriction to a general curve
often avoids the most pathological subvarieties of
$\overline{\mathcal{M}}_{g,n}(X,\beta)$, and therefore, the fixed-domain
Gromov-Witten invariants are much more often enumerative [16]. Furthermore,
even when enumerativity fails, the corresponding geometric count, which is
different, is usually well-defined and may still be accessible via more subtle
methods.

In this paper, we will restrict to the case in which the subvarieties
$X_{i}\subset X$ are points $x_{i}$, but many of our methods work more
generally. We now set up the problem precisely.

Fix integers $g,n\geq 0$ satisfying the stability condition $2g-2+n>0$ so that
the moduli space $\overline{\mathcal{M}}_{g,n}$ of $n$-pointed, genus $g$
stable curves is well-defined. Fix $\beta\in H_{2}(X,\mathbb{Z})$ an effective
curve class satisfying the condition

$\beta\cdot K_{X}^{\vee}>0.$

Let $\overline{\mathcal{M}}_{g,n}(X,\beta)$ be the moduli space of genus $g$,
$n$-pointed stable maps to $X$ in class $\beta$ and assume that the
dimensional constraint

$\mathrm{vdim}(\overline{\mathcal{M}}_{g,n}(X,\beta))=\mathrm{dim}(\overline{\mathcal{M}}_{g,n}\times
X^{n})$

holds. This is equivalent to

$\beta\cdot K_{X}^{\vee}=r(n+g-1).$ (1)

If the dimension constraint (1) holds, we expect a finite number of maps from
a fixed $n$-marked curve $(C,p_{1},\ldots,p_{n})$ of genus $g$ to $X$ in class
$\beta$ where the $p_{i}$ are incident to general points of $X$. Unless
otherwise specified, we will always assume that constraint (1) holds
throughout.

The virtual Tevelev degree $\mathsf{vTev}^{X}_{g,n,\beta}$ of $X$ is defined
in [4, Definition 1.1] to be the corresponding virtual count. Formally, let

$\tau:\overline{\mathcal{M}}_{g,n}(X,\beta)\rightarrow\overline{\mathcal{M}}_{g,n}\times
X^{n}$ (2)

be the canonical morphism obtained from the domain curve and the evaluation
maps. Then, $\mathsf{vTev}^{X}_{g,n,\beta}\in\mathbb{Q}$ is defined by the
equality

$\tau_{*}[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}=\mathsf{vTev}^{X}_{g,n,\beta}\cdot[\overline{\mathcal{M}}_{g,n}\times
X^{n}]\in A^{0}(\overline{\mathcal{M}}\times X^{n})_{\mathbb{Q}}.$

The geometric Tevelev degree $\mathsf{Tev}^{X}_{g,n,\beta}\in\mathbb{Z}$ of
$X$ is defined in [16, Definition 2] under the further assumption that the
restriction of $\tau$ to maps with smooth domain

$\tau:{\mathcal{M}}_{g,n}(X,\beta)\rightarrow{\mathcal{M}}_{g,n}\times X^{n}$

has reduced and $0$-dimensional general fiber, in which case its cardinality
is by definition $\mathsf{Tev}^{X}_{g,n,\beta}$.

We record here the following crucial lemma, see [16, Proposition 14, proof of
Proposition 22]. It does not require the assmption (1).

######  Lemma 1.

Let $f\subset{\mathcal{M}}_{g,n}(X,\beta)$ be an irreducible component.
Suppose that $Z$ dominates $X^{n}$ and that $n\geq g+1$. Then, $Z$ is
generically smooth of the expected dimension.

More generally, let $Z\subset\overline{\mathcal{M}}_{g,n}(X,\beta)$ be an
irreducible component whose general point $f:C\to X$ has the property that $C$
is a union of a smooth genus $g$ curve $C_{{\text{sp}}}$ containing $m$ of the
marked points, and $n-m$ rational tails attached to $C_{{\text{sp}}}$, each
containing a marked point. Suppose that $Z$ dominates $X^{n}$ and that $m\geq
g+1$. Then, $Z$ is generically smooth of the expected dimension.

In particular, $\mathsf{Tev}^{X}_{g,n,\beta}$ is well-defined whenever $n\geq
g+1$. This bound is not sharp: for example, when $X=\mathbb{P}^{r}$, the
geometric Tevelev degrees are also well-defined whenever $n\geq r+1$ (in which
case $f:C\to\mathbb{P}^{r}$ is necessary non-degenerate) by Brill-Noether
theory. It is an interesting question to determine for other $X$ whether the
bound $n\geq g+1$ can be improved.

Three questions emerge:

  1. 1.

What is $\mathsf{vTev}^{X}_{g,n,\beta}$?

  2. 2.

What is $\mathsf{Tev}^{X}_{g,n,\beta}$?

  3. 3.

Is $\mathsf{vTev}^{X}_{g,n,\beta}=\mathsf{Tev}^{X}_{g,n,\beta}$? (That is, is
$\mathsf{vTev}^{X}_{g,n,\beta}$ enumerative?)

The three questions are clearly not independent, but in practice involve
somewhat different ideas interacting in subtle ways.

This paper is concerned with aspects of all three questions when $X$ is a
blow-up of $\mathbb{P}^{r}$ at a collection of points. Other examples of
interest include complete intersections and homogeneous spaces [16, 4, 5, 14].
We next review the preliminary case $X=\mathbb{P}^{r}$.

###  1.2 Curve counts on $\mathbb{P}^{r}$

The answers to all three questions above on fixed-domain curve-counts are
fully understood for $X=\mathbb{P}^{r}$.

Virtual counts for $\mathbb{P}^{r}$ are easy to obtain. We write $\beta=d$ for
the class of $d$ times a line throughout this section.

######  Theorem 2.

[4, (3)] Assume (1). Independently of $d$, we have

${\mathsf{vTev}}^{\mathbb{P}^{r}}_{g,n,d}=(r+1)^{g}.$

In fact, this formula had been obtained much earlier [3] before the modern
theory of the Gromov-Witten virtual fundamental class was available.

The virtual and geometric counts agree when $d$ is large compared to $r$ and
$g$:

######  Theorem 3.

[9] Assume (1) and that $d\geq rg+r$ (equivalently, $n\geq d+2$). Then,

${\mathsf{Tev}}^{\mathbb{P}^{r}}_{g,n,d}=(r+1)^{g}.$

Obtaining geometric counts in _low_ degree for $\mathbb{P}^{r}$ is much more
difficult; the virtual and geometric counts no longer agree. For
$\mathbb{P}^{1}$, we have:

######  Theorem 4.

[7, 9, 6] Assume (1) for $r=1$. Then,

$\displaystyle{\mathsf{Tev}}^{\mathbb{P}^{1}}_{g,n,d}$
$\displaystyle=2^{g}-\sum_{j=0}^{g-d-1}\binom{g}{j}+(g-d-1)\binom{g}{g-d}+(d-g-1)\binom{g}{g-d+1}$
$\displaystyle=\int_{\mathrm{Gr}(2,d+1)}\sigma^{g}_{1}\cdot\sum_{a_{0}+a_{1}=n-3}\sigma_{a_{0}}\sigma_{a_{1}}$

Binomial coefficients $\binom{g}{j}$ with $j<0$ are interpreted to vanish,
recovering Theorem 3 when $d\geq r+1$.

Finally, the computation of geometric counts
${\mathsf{Tev}}^{\mathbb{P}^{r}}_{g,n,d}$ is completed in [15], given in terms
of Schubert calculus and a formula of Klyachko [12]. We do note restate the
formula here.

###  1.3 New results

In this paper, we study Tevelev degrees of blow-ups of projective spaces at
(general) points. We obtain new results on all three aspects of the question:
enumerativity, geometric, and virtual calculations.

####  1.3.1 Enumerativity

Fixed-domain virtual curve counts are much more likely to be enumerative than
arbitrary Gromov-Witten invariants. For example, we have already seen that
virtual and geometric counts for $\mathbb{P}^{r}$ agree in large degree,
whereas higher-genus Gromov-Witten invariants of $\mathbb{P}^{r}$ typically
fail to be enumerative. This phenomenon was first studied systematically in
[16].

We assume throughout that the dimensional constraint (1) holds.

######  Definition 5.

We say that $X$ satisfies strong asymptotic enumerativity (SAE) if :

  1. 1.

For all $g\geq 0$, there exists a constant $C(X,g)$ (depending only on $X$ and
$g$) for which, if $n\geq C(X,g)$, then the general fiber of the map (2)

$\tau:\overline{\mathcal{M}}_{g,n}(X,\beta)\rightarrow\overline{\mathcal{M}}_{g,n}\times
X^{n}$

is contained in ${\mathcal{M}}_{g,n}(X,\beta)$, and

  2. 2.

one can take $C(X,0)=3$.

If $\tau$ has general fiber contained in ${\mathcal{M}}_{g,n}(X,\beta)$ and
furthermore $n\geq g+1$, then by Lemma 1, the general fiber of $\tau$ is
reduced and $0$-dimensional, so
$\mathsf{Tev}^{X}_{g,n,\beta}=\mathsf{vTev}^{X}_{g,n,\beta}$, i.e.
$\mathsf{vTev}^{X}_{g,n,\beta}$ is enumerative. Thus, we have:

######  Proposition 6.

Suppose that $X$ satisfies SAE. Then, virtual Tevelev degrees are enumerative:

  * •

in arbitrary genus whenever $n$ (equivalently, $\beta\cdot K_{X}^{\vee}$) is
sufficiently large (depending on $g$), and

  * •

_always_ when $g=0$.

In [16], SAE is proven for homogeneous spaces for linear algebraic groups (see
[16, Theorem 10]) and non-singular hypersurfaces of very low degree (see [16,
Theorem 11, Corollary 34]), implying the conclusion of Proposition 6 in these
examples. As it is difficult to imagine how one could obtain this conclusion
for $X$ without SAE, the following question is natural.

######  Question 7.

What geometric conditions on (non-singular, projective) $X$ guarantee SAE?

It was originally expected that all _Fano_ $X$ satisfy SAE [16, Speculation 12
and §4]. However, we have been informed that examples of special Fano
hypersurfaces of large degree failing to satisfy SAE have been constructed in
forthcoming work of Beheshti-Lehmann-Riedl-Starr-Tanimoto. The SAE property
therefore seems somewhat more subtle.

Our first main set of results concerns SAE for $X$ given by the blowup of
$\mathbb{P}^{r}$ at distinct points $q_{1},\ldots,q_{\ell}$. We have the
following negative result:

######  Theorem 8.

Suppose that $r\geq 4$ and $\ell\geq 2$. Then, $X$ fails to satisfy SAE.

Such $X$ are not Fano, and indeed, the existence of negative rational curves
is used an in essential way. In fact, we prove a more general criterion for
failure of SAE which also applies to various special configurations of points
$q_{j}$ (e.g., where some three of the $q_{j}$ are collinear), see Proposition
15.

On the other hand, we have the following positive result.

######  Theorem 9.

$X={\mathsf{Bl}}_{q_{1},\ldots,q_{\ell}}(\mathbb{P}^{r})$ satisfies SAE in the
following cases:

  1. 1.

(Theorem 24) $X$ is del Pezzo, i.e. $r=2$, and the $\ell\leq 8$ points satisfy
the property that no three lie on a line, no six lie on a conic, and, if
$\ell=8$, the points do not all lie on a cubic singular at one of the $q_{j}$.

  2. 2.

(Theorem 32) $r=3,\ell\leq 4$, and the $q_{j}$ are (linearly) general.

  3. 3.

(Theorem 23) $r$ is arbitrary and $\ell=1$.

In Theorem 9, examples 1 and 3 are Fano; in fact, these are all examples of
Fano varieties obtained by blowing up $\mathbb{P}^{r}$ at general points. In
example 2, $X$ is merely $(-K_{X})$-nef when $\ell\geq 2$; these are the first
examples of non-Fano varieties proven to satisfy SAE.

When the points $q_{j}$ are general, the only cases left open are when
$r=2,\ell\geq 9$ and $r=3,\ell\geq 5$. We believe SAE should hold in these
examples, but we do not see a way to obtain unconditional proofs. In dimension
2, one would at least need a bounded negativity statement for general blowups
of $\mathbb{P}^{2}$, if not the full power of the SHGH conjecture (see [10]
for a survey), and the situation in dimension 3 seems even more hopeless.

####  1.3.2 Geometric calculations

In §3.1, we prove an integral formula for the geometric Tevelev degrees of
blow-ups $\pi:X\rightarrow\mathbb{P}^{r}$ at $\ell\leq r+1$ general points
$q_{1},\ldots,q_{\ell}$. Without loss of generality, we will take the $q_{j}$
to be fixed points under the standard action of a $(r+1)$-dimensional torus on
$\mathbb{P}^{r}$.

Let $\mathsf{H}\in H^{2}(X)$ denote the pullback of the hyperplane class from
$\mathbb{P}^{r}$, and let $\mathsf{E}_{1},\ldots,\mathsf{E}_{\ell}\in
H^{2}(X)$ be the classes of the exceptional divisors. Let
$\mathsf{H}^{\vee},\mathsf{E}^{\vee}_{1},\ldots,E^{\vee}_{\ell}\in H_{2}(X)$
be the corresponding dual basis, and let

$\beta=d\mathsf{H}^{\vee}+\sum_{i=1}^{\ell}k_{i}\mathsf{E}_{i}^{\vee}\in
H_{2}(X,\mathbb{Z})$

be an effective curve class.

We are interested in counting maps from a fixed general curve
$(C,p_{1},\ldots,p_{n})$ to $X$ in class $\beta$. Equivalently, we count maps
to $\mathbb{P}^{r}$ of degree $d$ and with multiplicities
$k_{1},\ldots,k_{\ell}$ at the points $q_{1},\ldots,q_{\ell}$, respectively.
Such a map is defined by a line bundle $\mathcal{L}$ of degree $d$ and $\ell$
divisors $D_{1},\ldots,D_{\ell}$ mapping to $q_{\ell}$. When $\ell\leq r+1$,
this is equivalent to the data of $r+1$ sections

$f_{i}\in H^{0}\left(C,\mathcal{L}\left(-\sum_{j\neq i}D_{j}\right)\right)$

which have no common vanishing point _viewed as sections of
$H^{0}(C,\mathcal{L})$._ We will parametrize such data in what follows.

Setup Write
$S:=\mathsf{Jac^{d}}(C)\times\mathsf{Sym}^{k_{1}}(C)\times\ldots\times\mathsf{Sym}^{k_{\ell}}(C)$.
Over $S\times C$, we have the following line bundles:

  1. $\bullet$

the pullback $\mathcal{P}$ from $\mathsf{Jac^{d}}(C)\times C$ of the Poincarè
line bundle;

  2. $\bullet$

for all $i=1,\ldots.,\ell$, the pullback $\mathcal{D}_{i}$ from
$\mathsf{Sym}^{k_{i}}(C)\times C$ of the universal divisor.

Let $\nu:S\times C\rightarrow S$ be the projection and define

$\mathcal{E}:=\bigoplus_{i=1}^{r+1}\nu_{*}\left(\mathcal{P}\left(-\sum_{j\neq
i}\mathcal{D}_{j}\right)\right).$

We will assume that

$d-\sum_{i\in I}k_{i}>2g-1\text{ for all }I\subseteq\\{1,\ldots,\ell\\}\text{
such that }|I|\leq r,$ (3)

in order to ensure that $\mathcal{E}$ is a vector bundle. We will see later in
§3.1 that we will also want this to remain true upon replacing $\mathcal{L}$
with $\mathcal{L}(-p_{i})$, hence the appearance of $2g-1$ rather than $2g-2$
on the right hand side.

Let $\eta:\mathbb{P}:=\mathbb{P}(\mathcal{E})\rightarrow S$ be the
corresponding projective bundle corresponding to _lines_ (not 1-dimensional
quotients) in the fibers of $\mathcal{E}$. Therefore, a point of $\mathbb{P}$
consists of the data of a point of $S$ and sections $f_{j}$ as above, taken up
to simultaneous scaling.

Our first result expresses Tevelev degrees of $X$ as integrals on
$\mathbb{P}$. Before stating the result, we require some additional notation.
We set

$\widetilde{\mathsf{H}}=c_{1}(\mathcal{O}_{\mathbb{P}}(1))-\eta_{1}-\cdots-\eta_{\ell}\in
H^{*}(\mathbb{P},\mathbb{Z})$ (4)

where $\eta_{i}$ is the pullback from $\mathsf{Sym}^{k_{i}}(C)$ of the class
of the divisor $N_{i}=\\{D:D-p\geq 0\\}$ (here $p$ is any fixed point in $C$).

Then, we have:

######  Proposition 10 (Integral formula).

Assume conditions (1), (3) hold, and in addition that

$n-d\geq g+1.$ (5)

Then, $\mathsf{Tev}_{g,n,\beta}$ is well-defined and the following integral
formula holds:

$\mathsf{Tev}_{g,n,\beta}^{X}=\int_{\mathbb{P}}(\widetilde{\mathsf{H}}^{r}+\sigma_{1}\widetilde{\mathsf{H}}^{r-1}+\cdots+\sigma_{r})^{n},$
(6)

where $\sigma_{i}$ is the $i$-th elementary symmetric function in
$\eta_{1},\ldots,\eta_{\ell}$ (so $\sigma_{i}=0$ for $i>\ell$).

The condition $n-d\geq g+1$ is needed to rule out extraneous contributions
when the sections $f_{j}$ are identically zero (see Lemma 40 and Remark 41),
and also ensure dimensional transversality later (Lemma 50). The factor
$\widetilde{\mathsf{H}}^{r}+\sigma_{1}\widetilde{\mathsf{H}}^{r-1}+\cdots+\sigma_{r}$
is the class of a subscheme $V(x_{i})$ of $\mathbb{P}$ corresponding to the
condition that $f(p_{i})=x_{i}$. However, care is needed to deal with the
possibility that the sections $f_{j}$ vanish simultaneously at $p_{i}$; a
precise definition of $V(x_{i})$ is given in §3.1. The computation of the
class $[V(x_{i})]$ and the fact that the $V(x_{i})$ intersect transversely at
points enumerated by $\mathsf{Tev}_{g,n,\beta}^{X}$ are both subtle.

Pushing forward to
$S=\mathsf{Jac^{d}}(C)\times\mathsf{Sym}^{k_{1}}(C)\times\cdots\times\mathsf{Sym}^{k_{\ell}}(C)$
using Grothendieck-Riemann-Roch, we obtain (see §3.2.1 for notation):

######  Theorem 11.

Assume conditions (1), 3 and 5 hold. Then,

$\mathsf{Tev}_{g,n,\beta}^{X}=\sum_{m=0}^{\min(n,k_{1},\ldots,k_{r+1})}\binom{n}{m}(-1)^{m}\int_{S}\prod_{i=1}^{r+1}(1+\eta_{i})^{n-m-1+g-d+\overline{k}_{i}}\eta_{i}^{m}\cdot\mathrm{exp}\left(\frac{\overline{\tau}_{i}+\Theta-\overline{x}_{i}}{1+\eta_{i}}\right).$

We extract from Theorem 11 relatively simple formulas in two special cases.

######  Theorem 12 ($g=0$ specialization).

Let $X$ be the blow-up of $\mathbb{P}^{r}$ at $\ell\leq r+1$ points and let

$\beta=d\mathsf{H}^{\vee}+\sum_{i=1}^{\ell}k_{i}\mathsf{E}_{i}^{\vee}\in
H_{2}(X,\mathbb{Z})$

be an effective curve class. Assume conditions (1), (3), and (5). Then,

$\mathsf{Tev}_{0,n,\beta}^{X}=\sum_{m=0}^{\min(k_{1},\ldots,k_{r+1},n)}(-1)^{m}\binom{n}{m}\prod_{i=1}^{r+1}\binom{n-d+\sum_{j\neq
i}k_{j}-1-m}{k_{i}-m}$

where we set $k_{\ell+1}=\cdots=k_{r+1}=0$ when $\ell<r+1$.

In particular, when $\ell<r+1$, the summation goes away, and
$\mathsf{Tev}_{0,n,\beta}^{X}$ is a product of binomial coefficients.

It is often the case that the formula above gives zero. For example, take
$\ell=1$ and $\beta=d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}$ (where we write
simply $\mathsf{E}_{1}^{\vee}=\mathsf{E}^{\vee}$), where $(r-1)k\leq d$.
Assuming (1), the inequalities (3), and (5) are satisfied. Then, Theorem 12
reads simply

$\mathsf{Tev}_{0,n,\beta}^{X}=\binom{n-d-1}{k}.$

If we assume further that $(r-1)k\leq d<(2r-1)k$, then we have $0\leq
n-d-1<k$, so we conclude that $\mathsf{Tev}_{0,n,\beta}^{X}=0$. In other
words, the map

$\tau:{\mathcal{M}}_{g,n}(X,\beta)\rightarrow{\mathcal{M}}_{g,n}\times X^{n}$

fails to be dominant. We do not have a geometric explanation for this
phenomenon.

Specializing Theorem 11 to $\ell=1$ in arbitrary genus yields:

######  Theorem 13 ($\ell=1$ specialization).

Let $X={\mathsf{Bl}}_{q}(\mathbb{P}^{r})$ and let

$\beta=d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}\in H_{2}(X,\mathbb{Z})$

be an effective curve class. Assume that conditions (1), (3), and (5) hold.
Then,

$\mathsf{Tev}_{g,n,\beta}^{X}=\sum_{m=0}^{g}(2r)^{g-m}(1-r)^{m}\binom{g}{m}\binom{n-d+g-m-1}{k}.$

Obtaining explicit formulas beyond these two specializations is cumbersome in
general. For example, we have checked that, if $X$ is the blow-up of
$\mathbb{P}^{2}$ at two points and $\beta\cdot K_{X}^{\vee}$ is sufficiently
large, then ${\mathsf{Tev}}^{X}_{g,n,\beta}$ is equal to

$\displaystyle\sum_{\begin{subarray}{c}a_{1}+b_{1}+b_{2}+b_{3}+a_{3}+a_{4}=g\\\
b_{1}+b_{2}+b_{3}=a_{2}\end{subarray}}\binom{g}{a_{1}\ a_{2}\ a_{3}\
a_{4}}\binom{a_{2}}{b_{1}\ b_{2}\
b_{3}}5^{a_{1}}(-1)^{b_{1}+a_{3}+a_{4}}\sum_{\ell=0}^{\mathrm{min}(k_{1}+a_{3}-a_{2},k_{2}+a_{4}-a_{2},b_{1})}(-1)^{\ell}\binom{b_{1}}{\ell}\cdot$
$\displaystyle\binom{2d-(k_{1}-a_{3})-g-n+1-b_{2}-b_{3}-\ell}{k_{1}-a_{2}-b_{2}-\ell}\cdot\binom{2d-(k_{2}-a_{4})-g-n+1-b_{2}-b_{3}-\ell}{k_{2}-a_{2}-b_{3}-\ell}.$

Note that all of our geometric results require the inequalities (3) and (5).
If $g$ and $k_{j}$ are viewed as fixed, then both inequalities hold for
sufficiently large $d$, so our results may be viewed as “large degree” counts
in analogy with Theorem 3. The low degree counts seem, unsurprisingly, more
difficult, and are left open.

We remark that when $d-(k_{1}+\cdots+k_{r})<0$ and $n\geq 1$, a map $f:C\to X$
as above would need to have image contained in the strict transform of the
torus-invariant hyperplane where the last coordinate is zero. On the other
hand, as the $x_{i}$ are general, they may be chosen not to lie on this
hyperplane, so it follows immediately that $\mathsf{Tev}_{g,n,\beta}^{X}=0$ in
this case.

####  1.3.3 Virtual calculations

Via a computation in the quantum cohomology ring $QH^{*}(X)$, we also obtain:

######  Theorem 14.

Let $X={\mathsf{Bl}}_{q}(\mathbb{P}^{r})$ and let

$\beta=d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}\in H_{2}(X,\mathbb{Z})$

be an effective curve class. Assume that (1) holds and that $n-d\geq 1$. Then,

$\mathsf{vTev}_{g,n,\beta}^{X}=\sum_{m=0}^{g}(2r)^{g-m}(1-r)^{m}\binom{g}{m}\binom{n-d+g-m-1}{k}.$

Thus, the formula matches that of Theorem 13, but holds in a wider range.

###  1.4 Acknowledgments

Portions of this work were completed during the first author’s visits to HU
Berlin in November 2022 (during the workshop “Resonance varieties, topological
invariants of groups, moduli”) and February 2023 (during the Northern German
Algebraic Geometry Seminar) and the second author’s visits to ETH Zürich in
July 2022 (during the ICM satellite meeting, Algebraic Geometry and Number
Theory) and November 2022. We thank both institutions for their continued
support. We also thank Gavril Farkas, Zhuang He, Woonam Lim, Rahul
Pandharipande, Eric Riedl, and Johannes Schmitt for helpful discussions.

A.C. was supported by SNF-200020-182181. C.L. was supported by an NSF
postdoctoral fellowship, grant DMS-2001976, and the MATH+ incubator grant
“Tevelev degrees.”

##  2 Enumerativity

###  2.1 Failure of SAE

We give a general construction of $X$ failing SAE.

######  Proposition 15.

Let $\pi:X\rightarrow\mathbb{P}^{r}$ be the blow-up of $\mathbb{P}^{r}$ at any
set of points $q_{1},\ldots,q_{m}\in\mathbb{P}^{r}$. Suppose that there exists
a curve $\mathbb{P}^{1}\cong D\subset X$, such that either:

  1. 1.

$D\cdot K_{X}^{\vee}<0$, or

  2. 2.

$D\cdot K_{X}^{\vee}=0$, $r=2$, and $\pi(D)$ is not a point.

Then, $X$ fails to satisfy SAE.

###### Proof.

We will prove that for every $g$, there is a divergent sequence
$\\{n_{k}=n_{k}[g]\\}_{k}$ such the general fiber of the map $\tau$ contains
maps with singular domain for all $n_{k}$.

We start by assuming that $D$ has negative degree against $K_{X}^{\vee}$. Fix
$g\geq 0$, and for $d\geq 0$, set

$\beta[d]=d\mathsf{H}^{\vee}+rD\in H_{2}(X,\mathbb{Z}),$

where $\mathsf{H}^{\vee}$ is the pullback of the class of a line on
$\mathbb{P}^{r}$. Notice that $\beta[d]\cdot K_{X}^{\vee}$ diverges for large
$d$. Let $n\geq 0$ be such that equation (1) is satisfied with
$\beta=\beta[d]$, i.e.

$d(r+1)+r(D\cdot K_{X}^{\vee})=r(n+g-1);$

in order for this to be possible, we assume that $d$ is divisible by $r$.

Let

${\mathcal{M}}_{\Gamma}\subseteq\partial\overline{\mathcal{M}}_{g,n}(X,\beta)$

be the locally closed locus consisting of stable maps where the domain curve
consists of a genus $g$ smooth curve mapping to $X$ with class
$d\mathsf{H}^{\vee}$ and containing all $n$ of the markings, attached to a
smooth rational tail mapping to $X$ with class $rD$. We claim that
${\mathcal{M}}_{\Gamma}$ dominates $\overline{\mathcal{M}}_{g,n}\times X^{n}$,
so that SAE fails.

To see this, notice that $m=n-D\cdot K_{X}^{\vee}>n$ satisfies the dimensional
constraint (1) for $\mathbb{P}^{r}$ with curve class $d\mathsf{H}^{\vee}$.
Therefore, since $\mathsf{Tev}^{\mathbb{P}^{r}}_{g,m,dH^{\vee}}>0$ (for
example, we have $m\geq d+2$ for large $d$, so we can apply [9, Theorem 1.1]),
for general $(C,p_{1},\ldots,p_{m})\in{\mathcal{M}}_{g,m}$ and
$x_{1},\ldots,x_{m}\in\mathbb{P}^{r}$ there exists

$f:(C,p_{1},\ldots,p_{m})\rightarrow\mathbb{P}^{r}$

in class $d\mathsf{H}^{\vee}$ mapping $p_{i}$ to $x_{i}$ for all
$i=1,\ldots,m$.

We are also free to assume that $x_{n+1}\in\pi(D)$. From $f$, we construct a
new map

$\tilde{f}:(\tilde{C},p_{1},\ldots,p_{n})\rightarrow X$

where $\widetilde{C}$ is obtained from $C$ by attaching a $\mathbb{P}^{1}$ at
$p_{n+1}$ and forgetting $p_{i}$ for $i>n+1$. The restriction $\widetilde{f}$
to $C$ is the unique morphism such that $\pi\circ\widetilde{f}=f$, and the
restriction to $\mathbb{P}^{1}$ is a fixed degree $r$ cover of $D$. The map
$[\widetilde{f}]$ lies in ${\mathcal{M}}_{\Gamma}$, and maps to a general
point of $\overline{\mathcal{M}}_{g,n}\times X^{n}$, as needed.

The proof in the second case is analogous: we have $m=n$ above, so we cannot
add an $(n+1)$-st point on $C$ constrained to map to $D$, but note that
$f(C)\cap\pi(D)\neq\emptyset$ automatically, so we can construct a map
$\tilde{f}$ as before. ∎

###### Proof of Theorem 8.

Apply Proposition 15 with $D$ equal to the strict transform of the line
between any two of the blown up points. ∎

Similarly, any blow-up of $\mathbb{P}^{r}$ at a set containing three collinear
points fails to satisfy SAE.

###  2.2 Generalities for Fano varieties

We now collect general statements used to prove SAE for Fano varieties; in
this subsection, we will always assume that $X$ is Fano.

######  Definition 16.

Let $\beta\in H_{2}(X,\mathbb{Z})$ be an effective curve class (possibly 0).
We say that $\beta$ is ordinary if the evaluation map
${\mathcal{M}}_{0,1}(X,\beta)\rightarrow X$ is dominant.

Fix a genus $g\geq 0$ and a non-zero effective curve class $\beta$ on $X$.
Assume that

$\mathrm{vdim}(\overline{\mathcal{M}}_{g,n}(X,\beta))\leq\mathrm{dim}(\overline{\mathcal{M}}_{g,n}\times
X^{n}).$ (7)

(Note: this is weaker than (1).)

For integers $a\geq 0$ and $m\in[0,n]$, define

${\mathcal{M}}_{\Gamma}^{(m,a)}\subseteq\overline{\mathcal{M}}_{g,n}(X,\beta)$

be the locally closed locus parametrizing maps whose domain $C$ has the
following topological type. $C$ contains a smooth _spine_ component
$C_{{\text{sp}}}$ of genus $g$, containing the marked points
$p_{n-m+1},\ldots,p_{n}$. Attached to $C_{{\text{sp}}}$, we have trees of
rational curves $T^{\prime}_{1},\ldots,T^{\prime}_{a}$, as well as trees of
rational curves $T_{1},\ldots,T_{n-m}$, such that the tree $T_{i}$ contains
the marked point $p_{i}$ for $i=1,\ldots,n-m$. See also Figure 1. Here
$(n-m)+a\geq 0$; we assume further that if $(n-m)+a=0$, that is, if $C$ is
irreducible, then (7) is a _strict_ inequality.

$\bullet$$p_{n-m+1}$$\bullet$$p_{n}$$C_{{\text{sp}}}$$T_{1}^{\prime}$$T_{a}^{\prime}$$\bullet$$p_{1}$$T_{1}$$\bullet$$p_{n-m}$$T_{n-m}$
Figure 1: Topological type of the domain curves of points in
${\mathcal{M}}_{\Gamma}^{(m,a)}$

Up to permuting indices of the marked points, the
${\mathcal{M}}_{\Gamma}^{(m,a)}$ contain all of the boundary strata that
dominate $\overline{\mathcal{M}}_{g,n}$. Therefore, $X$ satisfies SAE if and
only if, given $g,n$, where either $g=0$ and $n\geq 1$, or $n$ is large
(depending on $X$ and $g$), the space ${\mathcal{M}}_{\Gamma}^{(m,a)}$ fails
to dominate $\overline{\mathcal{M}}_{g,n}\times X^{n}$ for all $m,a$.

Let $Z$ be an irreducible component of ${\mathcal{M}}_{\Gamma}^{(m,a)}$ and
$[f]\in Z$ be a general point. For $n$ large (depending on $X$ and $g$) and
for $g=0$ and $n\geq 1$, consider the following two conditions:

  1. (*)

if $a=0$ and the pushforward under $f$ of each component of $T_{i}$ is an
ordinary class for $i=1,\ldots,n-m$, then we have the inequality

$\displaystyle\mathrm{dim}_{[f|_{C_{{\text{sp}}}}]}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))-\mathrm{vdim}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}$
$\displaystyle]))-n$ $\displaystyle\leq-(g+1);$

  2. (**)

for all $i=1,\ldots,n-m$, either $f_{*}[T_{i}]\cdot K_{X}^{\vee}\geq r+1$ or
the pushforward under $f$ of each component of $T_{i}$ is an ordinary class.

We will prove that if conditions (*) and (**) hold for all $Z$, then $X$
satisfies SAE. This will follow from the next two propositions.

######  Proposition 17.

Suppose that $a=0$, that, for $i=1,\ldots,n-m$, the pushforward under $f$ of
each component of $T_{i}$ is an ordinary class, and that condition (*) holds.
Then, $Z$ fails to dominate $\overline{\mathcal{M}}_{g,n}\times X^{n}$.

###### Proof.

Arguing as in [16, Proof of Proposition 22], we have

$\displaystyle\mathrm{dim}_{[f]}(Z)$
$\displaystyle\leq\mathrm{dim}_{[f|_{C_{{\text{sp}}}}]}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))-\mathrm{vdim}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))-n$
$\displaystyle+\mathrm{dim}(\overline{\mathcal{M}}_{g,n}\times X^{n})+m$

where the inequality is strict when $n=m$.

If $m\geq g+1$ and $Z$ dominates $\overline{\mathcal{M}}_{g,n}\times X^{n}$
then, by [16, Proposition 13], we have

$\mathrm{dim}_{[f|_{C_{{\text{sp}}}}]}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))=\mathrm{vdim}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))$

and this is a contradiction ($Z$ cannot dominate
$\overline{\mathcal{M}}_{g,n}\times X^{n}$ for dimensional reasons). When
instead $m<g+1$, then we conclude using Condition (*). ∎

######  Proposition 18.

Suppose that both conditions (*) and (**) hold. Then, $Z$ fails to dominate
${\mathcal{M}}_{g,n}\times X^{n}$.

###### Proof.

Because $X$ is Fano, we have

$f_{*}([T_{i}^{\prime}])\cdot K_{X}^{\vee}>0$

for all $i=1,\ldots,a$. Therefore, arguing as in [16, proof of Proposition
23], we may assume without loss of generality that $a=0$.

If the pushforward under $f$ of each irreducible component of each $T_{i}$ for
$i=1,\ldots,s$ is ordinary, then we are done by Proposition 17.

Suppose now without loss of generality that $T_{1},\ldots,T_{s}$ with $0<s\leq
n-m$ are trees each containing some component whose pushforward via $f$ is not
ordinary. Then, by condition (**), we have

$\mathrm{deg}(f_{*}[T_{i}])\geq r+1$

for all $i=1,\ldots,s$. Let $\hat{f}:\hat{C}\rightarrow X$ be the stable map
obtained from $f$ by deleting $T_{1},\ldots,T_{s}$ and $x_{1},\ldots,x_{s}$
and let $\widehat{\beta}=\widehat{f}_{*}[\widehat{C}]$. Notice that
$[\widehat{f}]$ belongs to an irreducible component $\widehat{Z}$ of

${\mathcal{M}}_{\Gamma}^{(m,0)}\subset\
\overline{\mathcal{M}}_{g,n-s}(X,\widehat{\beta})$

It is enough to show that $\widehat{Z}$ does not dominate
$\overline{\mathcal{M}}_{g,n-s}\times X^{n-s}$. This will follow from
Proposition 17, once we notice the following two facts. First, from (7) and
condition (**) we have

$\widehat{\beta}\cdot K_{X}^{\vee}=\beta\cdot
K_{X}^{\vee}-\sum_{i=1}^{s}f_{*}[T_{i}]\cdot K_{X}^{\vee}\leq r(n-s+g-1)-s$

and so

$\mathrm{vdim}(\overline{\mathcal{M}}_{g,n-s}(X,\widehat{\beta}))<\mathrm{dim}(\overline{\mathcal{M}}_{g,n-s}\times
X^{n-s}).$

Second, by Equation 7 and Condition (**), we have

$s\leq\frac{\beta\cdot K_{X}^{\vee}}{r+1}=\frac{r}{r+1}\cdot
n+\frac{r}{r+1}\cdot(g-1)$

and so for large $n$ also $n-s$ is large and for $g=0$ and $n\geq 1$, also
$n-s\geq 1$.

∎

To summarize, we have shown that:

######  Proposition 19.

Suppose that every irreducible component
$Z\subset{\mathcal{M}}_{\Gamma}^{(m,a)}$ as above satisfies Conditions (*) and
(**) for large $n$ (depending on $X$ and $g$), and for all $n\geq 1$ when
$g=0$. Then, $X$ satisfies SAE.

###  2.3 Blow-ups of $\mathbb{P}^{r}$

Let $X$ be the blow-up of $\mathbb{P}^{r}$ at $\ell$ general points. The next
results will later be useful to deal with conditions (*) and (**).

######  Lemma 20.

Let

$\beta=d\mathsf{H}^{\vee}+\sum_{i=1}^{\ell}k_{i}\mathsf{E}_{i}^{\vee}$

be a non-zero effective curve in $H_{2}(X,\mathbb{Z})$. If $\beta$ is
ordinary, then

$d\geq\sum_{i\in I}k_{i}\text{ for all }I\subseteq\\{1,\ldots,\ell\\}\text{
with }|I|\leq r.$

###### Proof.

Fix $I\subseteq\\{1,\ldots,r+1\\}$ such that $|I|=r$. Let
$f:\mathbb{P}^{1}\rightarrow X$ be a curve in class $\beta$; because $\beta$
is ordinary, we may assume that the image of $f$ is not contained the strict
transform $\Lambda_{I}$ of the hyperplane in $\mathbb{P}^{r}$ generated by the
points $\pi(\mathsf{E}_{i})$ for $i\in I$. In particular, we have

$0\leq\beta\cdot\Lambda_{I}=d-\sum_{i\in I}k_{i}.$

as desired. ∎

######  Lemma 21.

Let $0\neq\beta\in H_{2}(X,\mathbb{Z})$ be an effective ordinary curve class.
Then, $\beta\cdot K_{X}^{\vee}\geq 2$.

###### Proof.

By [16, Proposition 13], every irreducible component of
$\overline{\mathcal{M}}_{0,1}(X,\beta)$ dominating $X$ has dimension equal to
$\beta\cdot K_{X}^{\vee}+r-2$, which must be greater than or equal to the
dimension of $X$. ∎

######  Lemma 22.

Assume that $\ell\leq r+1$. Let $f:C\rightarrow X$ be a map from a smooth
genus $g$ curve in class
$d\mathsf{H}^{\vee}+\sum_{i=1}^{\ell}k_{i}\mathsf{E}_{i}^{\vee}$ with either
$d>0$ or $d=k_{1}=\ldots=k_{\ell}=0$. Then, $h^{1}(C,f^{*}T_{X})\leq rg$.

###### Proof.

When $d=k_{1}=\ldots=k_{\ell}=0$, we have an equality. Assume that $d>0$.

If $\ell\leq r+1$, then

$H^{0}(X,T_{X})\otimes\mathcal{O}_{X}\rightarrow T_{X}$

is surjective on $X\smallsetminus\bigcup_{i=1}^{\ell}\mathsf{E}_{i}$, and so
also

$H^{0}(C,f^{*}T_{X})\otimes\mathcal{O}_{C}\rightarrow f^{*}T_{X}$

is surjective on $C\smallsetminus\bigcup_{i=1}^{\ell}f^{-1}(\mathsf{E}_{i})$,
which is non-empty. Choosing $r$ global sections of $H^{0}(C,f^{*}T_{X})$,
linearly independent upon restriction to some point of $C$, yields a morphism

$\mathcal{O}_{C}^{\oplus r}\rightarrow f^{*}T_{X}$

which is surjective outside finitely many points on $C$. In particular, the
induced map

$H^{1}(C,\mathcal{O}_{C})^{\oplus r}\twoheadrightarrow H^{1}(C,f^{*}T_{X})$

is surjective, and this gives the stated conclusion. ∎

###  2.4 SAE for ${\mathsf{Bl}}_{q}(\mathbb{P}^{r})$

Let $X={\mathsf{Bl}}_{q}(\mathbb{P}^{r})$ be the blow-up of $\mathbb{P}^{r}$
at one point. Write $\mathsf{H},\mathsf{E}$ for the divisor classes of the
pulled-back hyperplane class and the exceptional divisor, respectively, and
write $\mathsf{H}^{\vee},\mathsf{E}^{\vee}$ for the corresponding dual basis
of $H_{2}(X)$.

######  Theorem 23.

$X$ satisfies SAE.

###### Proof.

We use Proposition 19. Let $Z\subseteq{\mathcal{M}}_{\Gamma}^{(m,a)}$ be an
irreducible component dominating $\overline{\mathcal{M}}_{g,n}\times X^{n}$
and let $[f]\in Z$ be a general point.

  1. Step 1

We verify Condition (*). Assume that $k=0$ and that the pushforward under $f$
of each component of $T_{i}$ is an ordinary class for $i=1,\ldots,n-m$. Write
$\beta=d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}$. We distinguish two cases.

    1. Case 1

Suppose that $f_{*}[C_{{\text{sp}}}]=-k^{\prime}\mathsf{E}^{\vee}$, with
$k^{\prime}>0$. Notice that, since the pushforward under $f$ of each component
of $T_{i}$ is an ordinary class for $i=1,\ldots,n-m$, by Lemma 20, we must
have

$d-(k+k^{\prime})\geq 0.$ (8)

On the other hand, we have

$\overline{\mathcal{M}}_{g,n}(X,-k^{\prime}\mathsf{E}^{\vee})=\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r-1},k^{\prime}),$

and so by Lemma 22,

$\displaystyle\mathrm{dim}_{[f|_{C_{{\text{sp}}}}]}\overline{\mathcal{M}}_{g,n}(X,-k^{\prime}\mathsf{E}^{\vee})$
$\displaystyle\leq\mathrm{vdim}(\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r-1},k^{\prime}))+rg$
$\displaystyle=(r-4)(1-g)+k^{\prime}r+n+rg.$

Therefore,

$\displaystyle\mathrm{dim}_{[f|_{C_{{\text{sp}}}}]}$
$\displaystyle(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))-\mathrm{vdim}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))+(r-1)g$
$\displaystyle=g-1+k^{\prime}+(r-1)g$ $\displaystyle\leq g-1+d-k+(r-1)g$
$\displaystyle<n-(g+1),$

where the last equality holds for every $n\geq 1$ when $g=0$ and for $n$ large
when $g>0$. Here, in the last two inequalities, we have used (8) and the
dimensional constraint (7).

    2. Case 2

If instead
$f_{*}[C_{{\text{sp}}}]=d^{\prime}\mathsf{H}+k^{\prime}\mathsf{E}^{\vee}$ with
$d^{\prime}\geq k^{\prime}\geq 0$ and $d^{\prime}>0$ or
$d^{\prime}=k^{\prime}=0$, then Condition (*) follows from Lemma 22.

  2. Step 2

We verify Condition (**). If some tree $T_{i}$ for $i\in\\{1,\ldots,n-m\\}$
contains a component $T_{i}^{0}$ whose pushforward via $f$ is not ordinary,
then it must be

$f_{*}[T_{i}^{0}]=-k^{\prime}\mathsf{E}^{\vee}\text{ for some }k^{\prime}>0$

which has degree $k^{\prime}(r-1)\geq r-1$. Also, $T_{i}$ has to contain a
component $\overline{T_{i}}$ whose pushforward via $f$ is ordinary and which,
by Lemma 21, has degree at least $2$. Therefore, $f_{*}[T_{i}]$ has at least
degree $r+1$.

∎

###  2.5 SAE for del Pezzo surfaces

######  Theorem 24.

Let $X$ be a del Pezzo surface. Then, $X$ satisfies SAE.

###### Proof.

We will use Proposition 19 again. Let
$Z\subseteq{\mathcal{M}}_{\Gamma}^{(m,a)}$ be an irreducible component and
$[f]\in Z$ a general point.

  1. Step 1

First, we verify Condition (*). If $m\geq g+1$, then by [16, Proposition 13],
we have $h^{1}(C_{{\text{sp}}},f^{*}T_{X})=0$, and so

$\mathrm{dim}_{[f|_{C_{{\text{sp}}}}]}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))=\mathrm{vdim}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))$

and Condition (*) holds.

If instead $m<g+1$, then $n-m\geq n-g-1$. Observe that, for all
$i=1,\ldots,n-m$, the pushforward of some component $\overline{T_{i}}$ of
$T_{i}$ must be an ordinary class different from $0$. Therefore, by Lemma 21,
we have

$f_{*}([T_{i}])\cdot K_{X}^{\vee}\geq 2\text{ for }i=1,\ldots,n-m,$

and so

$f_{*}[C_{{\text{sp}}}]\cdot K_{X}^{\vee}\leq\beta\cdot
K_{X}^{\vee}-2(n-m)<2((g+1)+g-1),$ (9)

where in the second inequality we used (7). In particular, when $g=0$, the
right hand side is $0$, yielding a contradiction. Therefore, for $g=0$, it
must be that $m\geq 1=g+1$ and $h^{1}(C_{{\text{sp}}},f^{*}T_{X})=0$.

For $g$ arbitrary, (9) implies that $f_{*}[C_{{\text{sp}}}]$ has uniformly
bounded degree against $K_{X}^{\vee}$, so there are only finitely many
possibilities for $f_{*}[C_{{\text{sp}}}]$ by the polyhedrality of the
effective cone of $X$. Therefore, $h^{1}(C_{{\text{sp}}},f^{*}T_{X})$ is
bounded by a constant $b[X,g]\in\mathbb{Z}_{\geq 0}$ (depending only on $X$
and $g$), and we have

$\mathrm{dim}_{[f|_{C_{{\text{sp}}}}]}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))-\mathrm{vdim}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))\leq
b[X,g].$

This implies that Condition (*) holds.

  2. Step 2

We now deal with Condition (**). Let $T_{i}$ be a tree containing a component
whose pushforward under $f$ is not ordinary. It still must contain a component
$\overline{T_{i}}$ such that $f_{*}([\overline{T_{i}}])$ is an ordinary curve
(and so of degree at least $2$ by Lemma 21), and at least one more irreducible
component, of degree at least $1$ because $X$ is Fano. Therefore,
$\mathrm{deg}(f_{*}([T_{i}])\geq 3=r+1$ and we are done.

∎

###  2.6 SAE in dimension $3$

Let $\pi:X\rightarrow\mathbb{P}^{3}$ be the blow-up of $\mathbb{P}^{3}$ at
$\ell\leq 4$ general points. Write
$\mathsf{H},\mathsf{E}_{1},\ldots,\mathsf{E}_{\ell}$ for the divisor classes
of the pulled-back hyperplane class and the exceptional divisor, respectively,
and write
$\mathsf{H}^{\vee},\mathsf{E}^{\vee}_{1},\ldots,\mathsf{E}^{\vee}_{\ell}$ for
the corresponding dual basis of $H_{2}(X)$.

In this section, we prove that $X$ satisfies SAE (Theorem 32). Note that such
$X$ are not Fano; this is, to our knowledge, the first example of a non-Fano
variety whose Tevelev degrees are asymptotically enumerative.

####  2.6.1 Low degree effective curves in $\mathbb{P}^{3}$

We begin by recalling a result about the effective cone of $X$.

######  Lemma 25.

[8, Proposition 4.1] The effective cone of $X$ is linearly generated by
$1$-dimensional linear spaces in the exceptional divisors and the strict
transforms of $1$-dimensional linear subspaces of $\mathbb{P}^{3}$, possibly
passing through the $\ell$ blown up points.

######  Corollary 26.

The anticanonical class
$K_{X}^{\vee}=4\mathsf{H}-2(\mathsf{E}_{1}+\cdots+\mathsf{E}_{\ell})$ is nef.

######  Remark 27.

For any curve class $\beta$, the intersection number $\beta\cdot K_{X}^{\vee}$
is always even; we will use this fact later.

######  Lemma 28.

Let $C\subset X$ be an irreducible curve of geometric genus 0 and with non-
positive degree against $K_{X}^{\vee}$. Then, $[C]\in H_{2}(X,\mathbb{Z})$ is
the class of a the strict transform of a line through two of the $q_{j}$. (In
particular, $C$ has degree 0 against $K_{X}^{\vee}$.)

###### Proof.

Write

$[C]=d\mathsf{H}^{\vee}+\sum_{j=1}^{\ell}k_{j}\mathsf{E}_{j}^{\vee};$

we may assume $d,k_{1},\ldots,k_{\ell}\geq 0$.

Let $\mathbb{T}\subset\mathbb{P}^{3}$ be a torus acting on $\mathbb{P}^{3}$
for which the $q_{j}$ are torus-fixed points. We also denote by
$\mathbb{T}\subset X$ the pullback to $X$, so that $\mathbb{T}$ also acts on
$X$. We claim that every curve in class $[C]$ lies in the strict transform
$\Lambda$ of some torus-fixed hyperplane of $\mathbb{P}^{3}$. If not, then
$[C]$ would be ordinary, and so by Lemma 21, would have degree at least 2.

To see that $[C]$ would be ordinary, let $\eta:\widetilde{C}\rightarrow C$ the
normalization map and let $p\in\widetilde{C}$ be such that $\eta(p)$ does not
belong to the strict transform of any torus fixed hyperplane of
$\mathbb{P}^{3}$. Then, the morphism

$\displaystyle\mathbb{T}\times\widetilde{C}$ $\displaystyle\rightarrow X$
$\displaystyle(t,x)$ $\displaystyle\mapsto t\cdot\eta(x)$

would yield a dominant map

$\mathbb{T}\rightarrow{\mathcal{M}}_{0,1}(X,[C])\xrightarrow{\mathrm{ev}_{p}}X$

implying that $[C]$ is ordinary.

Notice that $\Lambda$ is isomorphic to the blow-up of $\mathbb{P}^{2}$ of at
most $3$ points, and $[C]$ may be regarded as pushed forward from a class
$[C]\in H_{2}(\Lambda,\mathbb{Z})$. Then, if $[C]$ is not the class of the
line through two of the $q_{j}$, then

$d\geq k_{i}+k_{j}\text{ for all }1\leq i<j\leq 3,$ (10)

(we take $k_{j}=0$ for $j>\ell$ if $\ell\leq 2$), whence

$\displaystyle 1$ $\displaystyle\geq K_{X}^{\vee}\cdot\beta$
$\displaystyle=4d-2(k_{1}+k_{2}+k_{3})$
$\displaystyle=4d-(k_{1}+k_{2})-(k_{1}+k_{3})-(k_{2}+k_{3})$
$\displaystyle\geq d.$

On the other hand, the degree against $K_{X}^{\vee}$ of the classes
$\mathsf{H}^{\vee}+\mathsf{E}_{i}^{\vee}$ and $-\mathsf{E}_{j}^{\vee}$ is $2$,
so we find no other possibilities for $\beta$. ∎

######  Lemma 29.

Let $\beta\in H_{2}(X,\mathbb{Z})$ be an effective ordinary curve class such
that $K_{X}^{\vee}\cdot C=2$. Then, $\beta=\mathsf{H}^{\vee}+E^{\vee}_{j}$ for
some $j\in\\{1,\ldots,\ell\\}$.

###### Proof.

Write

$\beta=d\mathsf{H}^{\vee}+\sum_{j=1}^{\ell}k_{j}\mathsf{E}_{j}^{\vee}.$

Then, by hypothesis,

$2=\beta\cdot K_{X}^{\vee}=4d-2(k_{1}+\cdots+k_{\ell})$

and, since $\beta$ is ordinary, by Lemma 20 we have

$d\geq\sum_{j\in J}k_{j}\text{ for all }J\subseteq\\{1,\ldots,\ell\\}\text{
with }|J|=3.$

Therefore, we have $4d\geq 3(k_{1}+\cdots+k_{\ell})$ and so

$2-(k_{1}+\cdots+k_{\ell})=4d-3(k_{1}+\cdots+k_{\ell})\geq 0$

and we obtain $k_{1}+\cdots+k_{\ell}\in\\{0,1,2\\}$, from which the lemma
follows easily. ∎

The previous lemmas motivate the following definition.

######  Definition 30.

Let $0\neq\beta\in H_{2}(X,\mathbb{Z})$ be an effective curve class. We say
that:

  1. $\bullet$

$\beta$ is exceptional if $\beta=-k\mathsf{E}_{j}^{\vee}$ for some $k>0$ and
$j\in\\{1,\ldots,\ell\\}$,

  2. $\bullet$

$\beta$ is a fixed line if it is a positive multiple of the strict transform
of a line in $\mathbb{P}^{3}$ through two of the blown-up points, and

  3. $\bullet$

$\beta$ is a special line if it is not a fixed line and is the strict
transform of a line in $\mathbb{P}^{3}$ through one of the blown-up points.

We remark that exceptional classes and fixed lines are not ordinary, but
special lines are.

####  2.6.2 Proof of SAE

Let $\pi:X\rightarrow\mathbb{P}^{3}$ be the blow-up at $\ell\leq 4$ points
$q_{1},\ldots,q_{\ell}$. Fix a genus $g\geq 0$ and a non-zero effective curve
class $\beta$. We will follow the setting of §2.2, but need to make suitable
modifications now that $X$ is not Fano.

Assume inequality (7) holds. For integers $a\geq 0$ and $s,m\geq 0$ such that
$s+m\leq n$, define

${\mathcal{M}}_{\Gamma}^{(s,m,a)}\subseteq\overline{\mathcal{M}}_{g,n}(X,\beta)$

to be the locally closed locus parametrizing maps whose domain $C$ has the
following topological type. $C$ contains a smooth _spine_ component
$C_{{\text{sp}}}$ of genus $g$, containing the marked points
$p_{n-m+1},\ldots,p_{n}$. Attached to $C_{{\text{sp}}}$, we have trees of
rational curves $T^{\prime}_{1},\ldots,T^{\prime}_{a}$, as well as trees of
rational curves $T_{1},\ldots,T_{n-m}$, such that the tree $T_{i}$ contains
the marked point $p_{i}$ for $i=1,\ldots,n-m$. Furthermore, the integer $s$ is
such that $T_{i}$ contains a non-ordinary component if and only if $i\leq s$.
See also Figure 2.

$\bullet$$p_{n-m+1}$$\bullet$$p_{n}$$C_{{\text{sp}}}$$T_{1}$$\bullet$$p_{1}$$T_{s}$$\bullet$$p_{s}$$\bullet$$p_{s+1}$$T_{s+1}$$\bullet$$p_{n-m}$$T_{n-m}$$T_{a}^{\prime}$$T_{1}^{\prime}$$a$
rational treeswithout markingson them$s$ marked rational trees with atleast
one non-ordinary component$n-m-s$ marked ratio- nal trees with onlyordinary
components Figure 2: Topological type of the domain curves of points in
${\mathcal{M}}_{\Gamma}^{(m,s,a)}$

. Again, we assume that if $(n-m)+a=0$, that is, if $C$ is irreducible, then
(7) is a _strict_ inequality.

Let $Z$ be an irreducible component of ${\mathcal{M}}_{\Gamma}^{(s,m,k)}$. We
will prove:

######  Proposition 31.

If either $n$ is large (depending on $X$ and $g$) or $g=0$ and $n\geq 1$, then
the component $Z$ fails to dominate $X^{n}\times\overline{\mathcal{M}}_{g,n}$.

As an immediate corollary, we obtain:

######  Theorem 32.

$X$ satisfies SAE.

We prove Proposition 31 by first making several reductions. Let $[f]\in Z$ be
a general point.

Reduction 1: _It is enough to prove Proposition 31 for $a=0$_.

###### Proof.

Because $f_{*}[T_{i}^{\prime}]\cdot K_{X}^{\vee}\geq 0$, this is clear when
$m<n$, as we may simply delete the components $T^{\prime}_{i}$, so assume that
$m=n$. For the same reason, we can also assume that $a=1$, that
$T_{1}^{\prime}$ is irreducible, and that $f_{*}[T_{1}^{\prime}]\cdot
K_{X}^{\vee}=0$. It follows from Lemma 28 and stability that

$f_{*}[T_{1}^{\prime}]=m(\mathsf{H}^{\vee}+\mathsf{E}_{j}^{\vee}+\mathsf{E}_{j^{\prime}}^{\vee})\text{
for some }m\in\mathbb{Z}_{>0}\text{ and }1\leq j<j^{\prime}\leq\ell.$

Let $\mathsf{L}$ be the line in $\mathbb{P}^{3}$ through $\pi(\mathsf{E}_{j})$
and $\pi(\mathsf{E}_{j^{\prime}})$, and $\rho:\widetilde{X}\rightarrow X$ be
the blow-up of $X$ along the strict transform of $\mathsf{L}$. Call
$\overline{\beta}=\beta-f_{*}[T_{1}^{\prime}]$ and let
$\overline{Z}\subset{\mathcal{M}}_{g,n}(X,\overline{\beta})$ be the
irreducible component to which $[f|_{C_{{\text{sp}}}}]$ belongs. From
$\overline{Z}$ we get an irreducible substack

$\widetilde{Z}\subseteq{\mathcal{M}}_{g,n}(\widetilde{X},\widetilde{\beta})$

dominating $\overline{Z}$. Here, $\widetilde{\beta}\in
H_{2}(\widetilde{X},\mathbb{Z})$ is an effective curve class with the property
that

$\widetilde{\beta}\cdot\mathsf{E}>0$

where $\mathsf{E}$ is the class of the exceptional divisor in $\widetilde{X}$.

If $Z$ dominates $X^{n}\times{\mathcal{M}}_{g,n}$, then $\widetilde{Z}$
dominates $\widetilde{X}^{n}\times{\mathcal{M}}_{g,n}$, and applying [16,
Proposition 13] twice, we obtain

$\mathrm{dim}(Z)=n+\beta\cdot K_{X}^{\vee}>n+\widetilde{\beta}\cdot
K_{\widetilde{X}}^{\vee}\geq\mathrm{dim}(\widetilde{Z}),$

contradicting the fact that $\widetilde{Z}$ dominates $Z$. ∎

Assume henceforth that $a=0$.

Reduction 2: _It is enough to prove Proposition 31 for $s=0$_.

###### Proof.

If, for some $i\in\\{1,\ldots,s\\}$, we have $f_{*}[T_{i}]\cdot
K_{X}^{\vee}\geq 4=\mathrm{dim}(X)+1$, then arguing as in the proof of
Proposition 18 above, we can delete $T_{i}$ from the domain.

Thus, we may assume that, for every $i=1,\ldots,s$, we have $f_{*}[T_{i}]\cdot
K_{X}^{\vee}\leq 2$ (recall that curves on $X$ always have even anti-canonical
degree). Let

$T_{i}=\bigcup_{j=0}^{k_{i}}T_{i}^{j}$ (11)

be the decomposition in irreducible components of $T_{i}$. Then, by Lemmas 28
and 29, $f_{*}[T_{i}^{j}]$ is a fixed line for all but one
$j\in\\{0,\ldots.,k_{i}\\}$, say except for $j=0$, and $f_{*}[T_{i}^{0}]$ is a
special line $\mathsf{H}^{\vee}+\mathsf{E}_{h}^{\vee}$ for some
$h\in\\{1,\ldots,\ell\\}$.

Now, for $[f]\in Z$ such that $f(p_{i})=x_{i}$, we need $x_{i}\in
f(T_{i}^{0})$. Thus, the image $f(T_{i}^{0})$ is the strict transform of the
unique line between $q_{h}$ and $x_{i}$, while for $j\geq 1$, the reduced
image $f(T_{i}^{j})^{\mathrm{red}}$ is either a point or the strict transform
of the unique line in $\mathbb{P}^{3}$ through two of the points
$q_{1},\ldots,q_{\ell}$. By assumption, there is at least a $j$ for which
$[f(T_{i}^{j})^{\mathrm{red}}]$ is a fixed line. However, for general $x_{i}$,
two such curves do not meet in $X$. Thus, $Z$ cannot dominate
$\overline{\mathcal{M}}_{g,n}\times X^{n}$ unless $s=0$. ∎

Assume henceforth that $a=s=0$.

Reduction 3: _We can assume that each $T_{i}$ is irreducible and that
$f_{*}[T_{i}]$ is a special line for $i=1,\ldots.,n-m$._

###### Proof.

If $f_{*}[T_{i}]\cdot K_{X}^{\vee}\geq 4$, then arguing as in the proof of
Proposition 18, we can delete $T_{i}$. By Lemma 21, we can assume that if (11)
is the decomposition in irreducible components of $T_{i}$, then $f(T_{i}^{j})$
is a point unless $j=0$, and $f(T_{i}^{0})$ is the strict transform of a line
through one of the points $q_{1},\ldots,q_{\ell}$. In particular, replacing
$T_{i}$ with $T_{i}^{0}$, it is enough to prove Proposition 31 when
$T_{i}=T_{i}^{0}$. ∎

Assume henceforth that each $T_{i}$ is irreducible and that $f_{*}[T_{i}]$ is
a special line for $i=1,\ldots.,n-m$.

Reduction 4: _It is enough to prove the following analogue of condition (*):_

  1. (*’)

_The following inequality holds:_

$\displaystyle\mathrm{dim}_{[f|_{C_{{\text{sp}}}}]}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))-\mathrm{vdim}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}$
$\displaystyle]))-n$ $\displaystyle\leq-(g+1);$

###### Proof.

The proof is the same as that of Proposition 17. ∎

###### Proof of Proposition 32.

It is enough to prove that condition (*’) holds whenever $n$ is large
(depending on $X$ and $g$) and whenever $g=0$ and $n\geq 1$. We distinguish
two cases.

  1. Case 1

Suppose that $f_{*}[C_{{\text{sp}}}]=-k^{\prime}\mathsf{E}_{h}^{\vee}$ with
$k^{\prime}>0$, for some $h\in\\{1,\ldots,\ell\\}$. Then, $f(T_{i})$ must be
the strict transform of some line in $\mathbb{P}^{3}$ through $q_{h}$, and
$C_{{\text{sp}}}$ cannot contain any marked point (i.e., $m=0$). In
particular, we have

$\beta=f_{*}[C]=n\mathsf{H}^{\vee}+(n-k^{\prime})\mathsf{E}_{h}^{\vee},$

and so by Equation (1),

$3(n+g-1)=\beta\cdot K_{X}^{\vee}=4n-2(n-k^{\prime})=2n+2k^{\prime},$

from which we deduce

$k^{\prime}=\frac{n}{2}+\frac{3}{2}(g-1).$

Proceeding as in Theorem 23, we find

$\displaystyle\mathrm{dim}_{[f|_{C_{{\text{sp}}}}]}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))-\mathrm{vdim}(\overline{\mathcal{M}}_{g,n}(X,f_{*}[C_{{\text{sp}}}]))$
$\displaystyle\leq 2g-1+k^{\prime}$
$\displaystyle\leq\frac{7}{2}g-\frac{5}{2}+\frac{n}{2}$

which implies the required inequality (*’).

  2. Case 2

If
$f_{*}[C_{{\text{sp}}}]=d^{\prime}\mathsf{H}^{\vee}+k_{1}^{\prime}\mathsf{E}_{1}^{\vee}+\cdots+k_{\ell}^{\prime}\mathsf{E}_{\ell}^{\vee}$
with $d^{\prime}>0$ or it is the $0$ class, then condition (*’) follows from
Lemma 22.

This concludes the proof. ∎

##  3 Geometric counts

###  3.1 Integral formula on $\mathbb{P}$

In this section, we prove Proposition 10. We first recall the setup of §1.3.2.

Let $\pi:X\rightarrow\mathbb{P}^{r}$ be the blow-up of $\mathbb{P}^{r}$ at
$\ell\leq r+1$ torus-fixed points, and let

$\beta=d\mathsf{H}^{\vee}+\sum_{i=1}^{\ell}k_{i}\mathsf{E}_{i}^{\vee}\in
H_{2}(X,\mathbb{Z})$

be an effective curve class.

Let
$S:=\mathsf{Jac^{d}}(C)\times\mathsf{Sym}^{k_{1}}(C)\times\ldots\times\mathsf{Sym}^{k_{\ell}}(C)$,
and let $\nu:\mathbb{P}\to S$ be the projective bundle parametrizing
$\ell$-tuples of sections

$f_{i}\in H^{0}\left(C,\mathcal{L}\left(-\sum_{j\neq i}D_{j}\right)\right)$

up to simultaneous scaling. We will write

$\mathcal{L}^{\prime}:=\mathcal{L}\left(-\sum_{j=1}^{\ell}D_{j}\right).$

Recall also the notation

$\widetilde{\mathsf{H}}=c_{1}(\mathcal{O}_{\mathbb{P}}(1))-\eta_{1}-\cdots-\eta_{\ell}\in
H^{*}(\mathbb{P},\mathbb{Z})$

where $\eta_{i}$ is the pullback from $\mathsf{Sym}^{k_{i}}(C)$ of the class
of the divisor $N_{i}=\\{D:D-p\geq 0\\}$ (here $p$ is any fixed point in $C$).

Fix general points $x_{1},\ldots,x_{n}\in X$. We can assume that $x_{i}$ lies
in the locus where $\pi$ is an isomorphism and identify
$x_{i}\in\mathbb{P}^{r}$, so that we can write

$x_{i}=[x_{i,1}:\cdots:x_{i,r+1}].$

We wish to cut out subschemes of $V(x_{i})\subset\mathbb{P}$ corresponding to
the conditions $f(p_{i})=x_{i}$, but care must be taken when the $p_{i}$
vanishes at some of the $f_{j}$.

For all $i=1,\ldots,n$ and $a\leq r$, let $V(x_{i})_{a}\subseteq\mathbb{P}$
denote the closed subscheme of points

$(\mathcal{L},D_{1},\ldots,D_{\ell},\\{f=[f_{1}:\cdots:f_{r+1}]\\})$

satisfying the equations

$x_{i,j}\cdot f_{h}(p_{i})-x_{i,h}\cdot f_{j}(p_{i})=0$ (12)

for all $h,j$ with $1\leq h<j\leq a+1$, where we interpret $f_{h}$ and $f_{j}$
as sections of $\mathcal{L}^{\prime}(D_{h}+D_{j})$ (upon twisting by $D_{j}$
and $D_{h}$, respectively), and then restrict to $p_{i}$.

Write $V(x_{i})=V(x_{i})_{r}$. (Note more generally that the locus
$V(x_{i})_{a}$ corresponds to the condition that $f(p_{i})$ lies in a general
linear space of codimension $a$.) and define now

$V(x_{1},\ldots,x_{n})=\bigcap_{i=1}^{n}V(x_{i}).$

Theorem 10 is implied by the following two statements.

######  Proposition 33.

For all $i=1,\ldots,n$, we have

$[V(x_{i})_{a}]=\widetilde{\mathsf{H}}^{a}+\sigma_{1}\widetilde{\mathsf{H}}^{a-1}+\cdots+\sigma_{a}\text{
in }H^{2a}(\mathbb{P}).$

where $\sigma_{i}=\sigma_{i}(\eta_{1},\ldots,\eta_{a+1})$ for $i=1,\ldots,a$.

######  Proposition 34 (Transversality).

The scheme $V(x_{1},\ldots,x_{n})$ is reduced and $0$-dimensional. Moreover,
its $\mathbb{C}$-points are in bijection with the set of maps enumerated in
$\mathsf{Tev}^{X}_{g,n,\beta}$. (In particular, the $f_{j}$, when regarded as
sections of $\mathcal{L}$, do not simultaneously vanish at any $p\in C$.)

The proof of transversality is deferred to the next section. Here, we prove
Proposition 33. Throughout the rest of this section, we work with a fixed $i$,
and assume that $x_{i}=[1:\cdots:1]$. Also, by setting $\eta_{i}=0$ for
$i>\ell$, we assume that $\ell=r+1$.

######  Lemma 35.

For all $a\leq r$, the subscheme $V(x_{i})_{a}\subset\mathbb{P}$ is
generically reduced and irreducible of codimension $a$.

###### Proof.

We first prove the lemma upon restriction to the open set
$\mathbb{P}_{0}\subset\mathbb{P}$ where $p_{i}\notin D_{v}$ for all
$v=1,2,\ldots,r+1$. In this case, the proposition is already true upon
restriction to the fibers of the projective bundle $\mathbb{P}$. Fix
$\mathcal{L}\in\mathsf{Jac^{d}}C$ and $D_{v}\in\mathsf{Sym}^{k_{v}}C$, and
write

$\mathbb{P}^{\prime}:=\mathbb{P}\left(\bigoplus_{t=1}^{r+1}H^{0}(C,\mathcal{L}^{\prime}(D_{t}))\right).$

Indeed, the restriction of $V(x_{i})_{a}$ to $\mathbb{P}^{\prime}$ is the
linear space cut out by the $a$ independent linear equations
$f_{1}(p_{i})=f_{v}(p_{i})$ for $v=2,3,\ldots,a+1$, where we may as well
consider $f_{1},\ldots,f_{a+1}$ all as sections of the same line bundle
$\mathcal{L}$.

We now claim that $\mathbb{P}_{0}\cap V(x_{i})_{a}$ is dense in
$V(x_{i})_{a}$, which will prove the lemma. We show that any point

$(\mathcal{L},D_{1},\ldots,D_{r+1},\\{f=[f_{1}:\cdots:f_{r+1}]\\})\in
V(x_{i})_{a}$

is a limit of points in $\mathbb{P}_{0}\cap V(x_{i})_{a}$.

Let $B$ be the spectrum of a discrete valuation ring, and let $\pi:C\times
B\to B$ be the projection. Let $\widetilde{D}_{1},\ldots,\widetilde{D}_{r+1}$
be divisors on $C\times B$ restricting to $D_{1},\ldots,D_{r+1}$ on the
special fiber, and such that
$p_{i}\notin\widetilde{D}_{1},\ldots,\widetilde{D}_{r+1}$ upon restriction to
the generic fiber. For each $t$, we construct a section

$\widetilde{f}_{t}\in H^{0}(C\times
B,\mathcal{L}^{\prime}(\widetilde{D}_{t}))\cong
H^{0}(B,\pi_{*}(\mathcal{L}^{\prime}(\widetilde{D}_{t})))$

restricting to $f_{t}$ on the special fiber, and for which the restrictions of
$\widetilde{f}_{t}$ to the $\\{p_{i}\\}\times B$ are all equal to each other,
when regarded as sections of $H^{0}(B,\mathcal{L}|_{p_{i}})$. By the
assumption that $p_{i}\notin\widetilde{D}_{1},\ldots,\widetilde{D}_{r+1}$ on
the generic fiber, the $\widetilde{f}_{t}$ and $\widetilde{D}_{t}$ define a
$K(B)$-point of $\mathbb{P}_{0}\cap V(x_{i})_{a}$, as needed.

The construction is as follows. First, consider the restriction map

$\gamma:H^{0}(C\times B,\mathcal{L}^{\prime}(\widetilde{D}_{1}))\to
H^{0}\left(C\times
B,\left.\mathcal{L}^{\prime}(\widetilde{D}_{1})\right|_{0}\right)\cong
H^{0}\left(C,\mathcal{L}^{\prime}(D_{1}))\right),$

where $0\in B$ is the closed point. We have $H^{1}(C\times
B,\mathcal{L}^{\prime}(\widetilde{D}_{1})(-C\times 0))=0$ by Cohomology and
Base Change and the Leray spectral sequence, so $\gamma$ is surjective, and we
may define $\widetilde{f}_{1}$ to be any section extending $f_{1}$.

Similarly, the restriction map

$\gamma^{\prime}:H^{0}(C\times B,\mathcal{L}^{\prime}(D_{1}))\to
H^{0}\left(C\times B,\left.\mathcal{L}^{\prime}(D_{1})\right|_{(C\times
0)\cup(p_{i}\times B)}\right)$

is surjective, because $H^{1}(C\times B,\mathcal{L}^{\prime}(D_{1})(-(C\times
0)-(p_{i}\times B)))=0$. Note here that in order to ensure that
$R^{1}\pi_{*}(\mathcal{L}^{\prime}(D_{1})(-(C\times 0)-(p_{i}\times B)))=0$,
we need the right hand side of (3) to be $2g-1$, rather than $2g-2$.
Therefore, we can define all other $\widetilde{f}_{t}$ extending $f_{t}$ _and_
with $\widetilde{f}_{1}(p_{i})=\widetilde{f}_{t}(p_{i})$ in $\mathcal{L}$. ∎

For all $t=1,2,\ldots,r+1$, let $W_{t}\subset\mathbb{P}$ denote the divisor
cut out by the equation $f_{t}(p_{i})=0$, where $f_{t}$ is viewed as a section
of $\mathcal{L}^{\prime}(D_{t})$. Similarly, for all $t=1,2,\ldots,r$, let
$W_{t,t+1}$ denote the locus on $\mathbb{P}$ of points satisfying
$f_{t}(p_{i})-f_{t+1}(p_{i})=0$, where $f_{t}-f_{t+1}$ is viewed as a section
of $\mathcal{L}^{\prime}(D_{t}+D_{t+1})$. Condition (3) implies that $W_{t}$
and $W_{t,t+1}$ are indeed divisors.

######  Lemma 36.

We have

$[W_{t}]=\widetilde{\mathsf{H}}+\eta_{a}$

and

$[W_{t,t+1}]=\widetilde{\mathsf{H}}+\eta_{a}+\eta_{a+1}$

in $H^{2}(\mathbb{P})$.

###### Proof.

We prove the second statement; the first is similar. The locus $W_{t,t+1}$ is
cut out by a tautological section of the line bundle

$\mathcal{O}_{\mathbb{P}}(1)\otimes\nu_{*}\left(\left.\mathcal{P}\left(-\sum_{v\neq
t,t+1}\mathcal{D}_{v}\right)\right|_{p_{i}}\right)\cong\mathcal{O}_{\mathbb{P}}(1)\otimes\bigotimes_{v\neq
t,t+1}\mathcal{O}_{\mathsf{Sym}^{k_{v}}C}(-N_{v}),$

where $\nu:C\times S\to S$ is the projection. Here, we have used the
isomorphisms

$\mathcal{P}|_{\mathsf{Jac^{d}}(C)\times\\{p_{i}\\}}\cong\mathcal{O}_{\mathsf{Jac^{d}}(C)}$

and

$\mathcal{O}(-\mathcal{D}_{v})|_{\mathsf{Sym}^{k_{v}}(C)\times\\{p_{i}\\}}\cong\mathcal{O}_{\mathsf{Sym}^{k_{v}}C}(-N_{v})\text{
for }v=1,...,r+1.$

Therefore, the desired class is

$\mathsf{H}-\sum_{v\neq
t,t+1}\eta_{v}=\widetilde{\mathsf{H}}+\eta_{t}+\eta_{t+1}\in
H^{2}(\mathbb{P}).$

∎

######  Lemma 37.

Let $[V]_{a}\in H^{2a}(\mathbb{P})$ be the class of
$V(x_{i})_{a}\subset\mathbb{P}$. Then, for all $a\leq r$, we have

$[V]_{a}=[V]_{a-1}(\widetilde{\mathsf{H}}+\eta_{a}+\eta_{a+1})-[V]_{a-2}(\widetilde{\mathsf{H}}+\eta_{a})\eta_{a},$
(13)

where we set $[V]_{-1}=0$ by convention.

###### Proof.

The claim follows from the following three statements, after applying Lemma
36.

  1. (1)

We have a set-theoretic equality

$(V_{a-1}\cap W_{a,a+1})=V_{a}\cup(V_{a-2}\cap W_{a}\cap N_{a})$

in $\mathbb{P}$.

  2. (2)

The subschemes $V_{a-1}\cap W_{a,a+1}$ and and $V_{a-2}\cap W_{a}\cap N_{a}$
of $\mathbb{P}$ (in addition to $V_{a}$) are generically reduced and
irreducible of codimension $a$.

  3. (3)

$V_{a-1}\cap W_{a,a+1}$ and $V_{a-2}\cap W_{a}\cap N_{a}$ are not equal.

Consider (1). The inclusion $\supset$ is straightforward. A point in $f\in
V_{a-1}\cap W_{a,a+1}$ satisfies (12) whenever $1\leq h<j\leq a$, and in
addition for $(h,j)=(a,a+1)$. If is not the case that $f\in V_{a}$, then (12)
must fail for $j=a+1$ and some $h<a$; without loss of generality, take $h=1$.

The sections $f_{1}\in\mathcal{L}^{\prime}(D_{1})$ and
$f_{a+1}\in\mathcal{L}^{\prime}(D_{a+1})$ are not equal at $p_{i}$ when
regarded as sections of $\mathcal{L}^{\prime}(D_{1}+D_{a+1})$, but _are_ equal
at $p_{i}$ when regarded as sections of
$\mathcal{L}^{\prime}(D_{1}+D_{a}+D_{a+1})$, by applying (12) for
$(h,j)=(1,a),(a,a+1)$. Therefore, we must have $p_{i}\in D_{a}$, that is,
$f\in N_{a}$.

Because we furthermore have $f\in W_{a,a+1}$, we conclude that $f_{a}$ is zero
at $p_{i}$ as a section of $\mathcal{L}^{\prime}(D_{a}+D_{a+1})$. Similarly,
applying (12) for $(h,j)=(1,a)$, we have that $f_{a}$ is zero at $p_{i}$ as a
section of $\mathcal{L}^{\prime}(D_{1}+D_{a})$. Now, either $f_{a}$ is zero as
a section of $\mathcal{L}^{\prime}(D_{a})$, in which case we are done, or
$f\in N_{1}\cap N_{a+1}$, in which case (12) holds for $(h,j)=(1,a+1)$,
contradicting the assumption at the beginning.

(2) is proven exactly as in Lemma 35, namely, $V_{a-1}\cap W_{a,a+1}$ is
generically a sub-projective bundle of $\mathbb{P}$ of codimension $a$, and
$V_{a-2}\cap W_{a}\cap N_{a}$ is generically a sub-projective bundle of the
pullback of $\mathbb{P}$ over $N_{a}$ of codimension $a-1$. The details are
omitted.

(3) is immediate, for example, from the fact that $N_{a}$ does not contain
$V_{a-1}\cap W_{a,a+1}$. ∎

###### Proof of Proposition 33.

We proceed by induction on $a$. When $a=1$, this is Lemma 36. Suppose $a>1$.
By the inductive hypothesis and Lemma 37, we have

$\displaystyle[V]_{a}=$
$\displaystyle[V]_{a-1}(\widetilde{\mathsf{H}}+\eta_{a}+\eta_{a+1})-[V]_{a-1}(\widetilde{\mathsf{H}}+\eta_{a})\eta_{a}$
$\displaystyle=$
$\displaystyle\left(\widetilde{\mathsf{H}}^{a-1}+\sigma_{1}(\eta_{1},\ldots,\eta_{a})\widetilde{\mathsf{H}}^{a-2}+\cdots+\sigma_{a-1}(\eta_{1},\ldots,\eta_{a})\right)(\widetilde{\mathsf{H}}+\eta_{a}+\eta_{a+1})$
$\displaystyle-\left(\widetilde{\mathsf{H}}^{a-2}+\sigma_{1}(\eta_{1},\ldots,\eta_{a-1})\widetilde{\mathsf{H}}^{a-3}+\cdots+\sigma_{a-2}(\eta_{1},\ldots,\eta_{a-1})\right)(\widetilde{\mathsf{H}}+\eta_{a})\eta_{a}$
$\displaystyle=$
$\displaystyle\widetilde{\mathsf{H}}^{a}+\sigma_{1}(\eta_{1},\ldots,\eta_{a+1})\widetilde{\mathsf{H}}^{a-1}+\cdots+\sigma_{a}(\eta_{1},\ldots,\eta_{a+1})$

where in the last equality we used the identities

$\displaystyle(\eta_{a}+\eta_{a+1})\sigma_{i-1}(\eta_{1},\ldots,\eta_{a})+\sigma_{i}(\eta_{1},$
$\displaystyle\ldots,\eta_{a})-\eta_{a}\sigma_{i-1}(\eta_{1},\ldots,\eta_{a-1})$
$\displaystyle-\eta_{a}^{2}\sigma_{i-2}(\eta_{1},\ldots,\eta_{a-1})=\sigma_{i}(\eta_{1},\ldots,\eta_{a+1})$

for $i=1,\ldots,a$. ∎

####  3.1.1 Interlude on the permutohedral variety

For the proof of transversality, it will be convenient for formal reasons to
pass from $X$ to a further blow-up $Y$, the permutohedral variety, at higher-
dimensional linear subspaces. We now describe this blow-up and its relevant
properties.

Fix a dimension $r\geq 2$. Write $[r+1]=\\{1,2,\ldots,r+1\\}$. For a non-empty
subset $S\subsetneq[r+1]$, let $\Lambda_{S}\subset\mathbb{P}^{r}$ denote the
torus-invariant linear space of dimension $(\\#S-1)$ given by the vanishing of
the coordinates indexed by the complement $[r+1]\backslash S$.

######  Definition 38.

Let $\rho:Y\to X$ be the blow-up of $X$ (which is itself obtained by blowing
up the torus-fixed points of $\mathbb{P}^{r}$) at the strict transforms of the
$\binom{r+1}{2}$ torus-invariant lines of $\mathbb{P}^{r}$, followed by the
strict transforms of the $\binom{r+1}{3}$ torus-invariant planes of
$\mathbb{P}^{r}$, and so on, through the torus-invariant codimension 2
subspaces.

######  Definition 39.

For $S\subset[r+1]$ of cardinality at most $r-1$, let $\mathsf{E}_{S}$ be the
class of the strict transform of $\Lambda_{S}$. Let
$\mathsf{H}^{\vee},\mathsf{E}_{S}^{\vee}\in H_{2}(Y)$ be the basis of 1-cycles
dual to the basis $\mathsf{H},\mathsf{E}_{S}\in H^{2}(Y)$ of divisors.

Note that

$K_{Y}=(r+1)\mathsf{H}-\sum_{\begin{subarray}{c}S\subset[r+1]\\\ 1\leq\\#S\leq
r-1\end{subarray}}(r-\\#S)\mathsf{E}_{S}.$

A point $y\in Y$ can be expressed in coordinates as follows. First, take a
point $y_{0}\in\mathbb{P}^{r}=\mathbb{P}(\mathbb{C}[[r+1]]),$ where
$\mathbb{C}[[r+1]]$ denotes the vector space with basis given by the set
$[r+1]=\\{1,2,\ldots,r+1\\}$. The point $y_{0}$ is the image of $y$ under the
composite blowup $Y\to\mathbb{P}^{r}$. Then, let $S^{1}\subsetneq[r+1]=:S^{0}$
be the subset of coordinates of $y_{0}$ which are equal to zero, corresponding
to the minimal $T$-invariant subvariety of $\mathbb{P}^{r}$ in which $y_{0}$
lies. Then, let
$y_{1}\in\mathbb{P}(\mathbb{C}[S^{1}])\cong\mathbb{P}^{\\#S^{1}-1}$ be a
point, representing a projectivized normal vector to $\Lambda_{(S^{1})^{c}}$
at $y_{0}$. Define $S^{2}\subsetneq S^{1}$ analogously, as the set of
coordinates of $y_{1}$ equal to zero, and continue until $y_{k}$ has all
coordinates non-zero. Then, $y$ consists of the data of the points
$(y_{0},y_{1},\ldots,y_{k})$.

Let $S\subset[r+1]$ be any non-empty subset. Then, we have a projection map
$\rho_{S}:Y\to\mathbb{P}(\mathbb{C}[S])$ given by remembering the coordinates
of $y=(y_{0},y_{1},\ldots,y_{k})$ corresponding to $S$, at the unique point
$y_{k^{\prime}}$ for which $S\subseteq S^{k^{\prime}}$ and the corresponding
coordinates are not all zero.

Let $C$ be a smooth curve. A map $f:C\to Y$, which, upon post-composition with
the blow-up $Y\to\mathbb{P}^{r}$, has image not contained in any torus-
invariant subvariety, may be is given by the following data:

  * •

a line bundle $\mathcal{L}$ on $C$,

  * •

for each $S\subset[r+1]$ with $1\leq\\#S\leq r-1$, an effective divisor
$D_{S}\subset C$, and

  * •

for each $j\in[r+1]$, a non-zero section

$f_{j}\in H^{0}\left(C,\mathcal{L}\left(-\sum_{j\notin S}D_{S}\right)\right),$

such that, when regarded as sections in $H^{0}(C,\mathcal{L})$, the $f_{j}$
have no common vanishing locus.

The point $f(p)=(y_{0},\ldots,y_{k})$ may be computed as follows. First
$y_{0}$ is the point $[f_{0}(p):\cdots:f_{r+1}(p)],$ where the $f_{j}$ are
regarded as sections of $\mathcal{L}$. Then, the subset $S^{1}\subset[r+1]$ is
the set of indices $j$ for which $f_{j}(p)=0$. The point $y_{1}$ has
coordinates given by the values of $f_{j}(p)$ after twisting $\mathcal{L}$
down by the unique positive multiple of $p$ for which the $f_{j}(p)$ are well-
defined and not all zero for $j\in S^{1}$. The rest of the
$y_{2},\ldots,y_{k}$ are then determined by further twists of $\mathcal{L}$.

####  3.1.2 Transversality

In this section, we prove Proposition 34. We continue to assume $\ell=r+1$.
The main difficulty is to show that a point of $V(x_{1},\ldots,x_{n})$
represents an “honest” map in class $\beta$. More precisely, let
$\mathbb{P}^{\circ}\subset\mathbb{P}$ be the open locus of
$f=[f_{1}:\cdots:f_{r+1}]$ for which the $f_{j}$ share no common zeroes, when
simultaneously viewed as sections of $\mathcal{L}$. Then, we wish to show that
$V(x_{1},\ldots,x_{n})\subset\mathbb{P}^{\circ}$.

In order for our argument to work, we will first need the following lemma.

######  Lemma 40.

Suppose $f=[f_{1}:\cdots:f_{r+1}]$ is a point of $V(x_{1},\ldots,x_{n})$.
Then, for each $j$, we have $f_{j}\neq 0$ as a section of
$\mathcal{L}^{\prime}(D_{j})$

###### Proof.

Suppose, without loss of generality, that $f_{1}=0$ and that $f_{2}\neq 0$ as
sections of $\mathcal{L}^{\prime}(D_{1})$ and $\mathcal{L}^{\prime}(D_{2})$,
respectively (note that we cannot have all $f_{j}=0$). We may assume that none
of the $x_{i}$ lie in the hyperplane in which $x_{i,1}=0$, so we must have
that $f_{2}$ vanishes at _all_ $p_{i}$ as a section of
$\mathcal{L}^{\prime}(D_{1}+D_{2})$. In particular, we need
$\deg(\mathcal{L}^{\prime}(D_{1}+D_{2}))\geq n$, that is,

$d-(k_{3}+\cdots+k_{r+1})\geq n,$

which immediately contradicts the assumption that $n-d\geq g+1$. ∎

######  Remark 41.

We will later use the assumption $n-d\geq g+1$ again (see Lemma 50), but we
point out that it is crucial for Lemma 40 to hold, which in turn is necessary
for the transversality.

For example, when $g=0$ and $r\geq 3$, suppose that the class
$\beta=d\mathsf{H}^{\vee}+k\mathsf{E}_{1}^{\vee}$ satisfies $0\leq k\leq
d\leq(r-1)k-r$. Then, we have $n\leq d$, so one can take
$f_{2}=\cdots=f_{r+1}=0$ and $f_{1}\in H^{0}(\mathbb{P}^{1},\mathcal{O}(d))$
to vanish at all of the $p_{1},\ldots,p_{n}$. This construction produces
points of $V(x_{1},\ldots,x_{n})$ not lying in $\mathbb{P}^{\circ}$, and
infinitely many such if $d>n$.

On the other hand, when $g=0$ and $r=2$, one can check that the assumption
$n-d\geq 1$ can be dropped, also later in Lemma 50.

We now turn to the main argument. The strategy is as follows: for any initial
$f_{\text{init}}\in V(x_{1},\ldots,x_{n})$, we describe an algorithm that
essentially amounts to twisting down at base-points of $f_{\text{init}}$ until
we are able to define a map $f:C\to\mathbb{P}^{r}$. The map $f$ will also have
various incidence conditions with respect to the torus-invariant loci of
$\mathbb{P}^{r}$ and the $p_{i},x_{i}$; a naive dimension count would
immediately predict the inexistence of such a map, unless
$f\in\mathbb{P}^{\circ}$ to begin with. We show that this expectation holds by
passing from $f$ to a map from $f:C\to Y$, where $Y$ is the permutohedral
variety of the previous section, moving in a family of the expected dimension
by Lemma 1.

Our procedure consists of a sequence of modifications to the following data:

  * •

A line bundle $\mathcal{L}$,

  * •

For every non-empty subset $S\subsetneq\\{1,2,\ldots,r+1\\}$, an effective
divisor $D_{S}\subset C$, and

  * •

For all $j=1,2,\ldots,r+1$, a _non-zero_ (see Lemma 40) section

$f_{j}\in H^{0}\left(C,\mathcal{L}\left(-\sum_{j\notin S}D_{S}\right)\right).$

The initial data is given by the data underlying the point
$f_{\text{init}}\in\mathbb{P}$: namely, the underlying line bundle denoted
$\mathcal{L}_{\text{init}}$, the divisors $D_{\\{j\\}}:=D_{j}$ and $D_{S}=0$
if $\\#S>1$, and the sections $(f_{j})_{\text{init}}$ of
$\mathcal{L}_{\text{init}}(-\sum_{v\neq j}D_{j})$. We say that $p$ is a _base-
point_ for the above data if $f_{j}(p)=0$ as a section of $\mathcal{L}$ for
all $j=1,2,\ldots,r+1$.

Throughout, we distinguish the initial line bundle and sections
$\mathcal{L}_{\text{init}},(f_{j})_{\text{init}}$, which do not change, from
the $\mathcal{L},f_{j}$, which do. The letter $f$ is only used at the end when
we obtain a map $f:C\to\mathbb{P}^{r}$ after no base-points remain. Similarly,
we use $D_{j}$ to denote the divisor associated to the point
$f_{\text{init}}\in V(x_{1},\ldots,x_{n})$, which does not change, to
distinguish it from $D_{\\{j\\}}$, which does.

We will repeatedly consider the following conditions on $f_{j}$ given $p\in
C$:

  1. (i)

$p\in D_{\\{j\\}}$

  2. (ii)

$f_{j}(p)=0$ as a section of $H^{0}\left(C,\mathcal{L}\left(-\sum_{v\neq
j}D_{\\{v\\}}\right)\right)$

We now describe the algorithm. Write $C^{\circ}=C-\\{p_{1},\ldots,p_{n}\\}$.
The first three steps will be carried out independently for all points $p\in
C^{\circ}$; we fix such a point $p$ throughout.

  1. Step 1

($p\in C^{\circ}$ satisfies (ii) for _all_ $j$)

Let $\alpha>0$ be the largest order to which $p$ vanishes at every $f_{j}$ (as
a section of $\mathcal{L}(-\sum_{v\neq j}D_{\\{v\\}})$) to order $\alpha$.
Then, make the following modifications:

     * •

$\mathcal{L}$ is replaced by $\mathcal{L}(-\alpha p)$,

     * •

each $D_{S}$ stays the same (in particular, all $D_{S}$ with $\\#S>1$ remain
empty), and

     * •

the new section $f_{j}\in H^{0}\left(C,\mathcal{L}\left(-\sum_{v\neq
j}D_{\\{v\\}}\right)\right)$ is taken to be the pre-image of the original
section $f_{j}$ under the inclusion

$H^{0}\left(C,\mathcal{L}(-\alpha p)\left(-\sum_{v\neq
j}D_{\\{v\\}}\right)\right)\to H^{0}\left(C,\mathcal{L}\left(-\sum_{v\neq
j}D_{\\{v\\}}\right)\right).$

(In other words, $f_{j}$ is “twisted down by $\alpha$ at $p$.”)

After step 1, (ii) may still hold for some $j$, but will no longer hold for
all $j$.

  2. Step 2

($p\in C^{\circ}$ satisfies (i) and (ii) for _some_ $j$)

For any $p,j$ as above, let $\alpha_{p,j}>0$ be the maximal order to which
both properties (i), (ii) hold, that is, $D_{\\{j\\}}$ contains $p$ with
multiplicity $\alpha_{p,j}$ and $f_{j}$ vanishes at $p$ to order
$\alpha_{p,j}$. Then, make the following modifications (independently for all
$j$):

     * •

$\mathcal{L}$ is replaced by $\mathcal{L}(-\alpha_{p,j}p)$,

     * •

$D_{\\{j\\}}$ is replaced by $D_{\\{j\\}}-\alpha_{p,j}p$, while all $D_{S}$
with $\\#S>1$ remain empty, and

     * •

$f_{j}$ is twisted down by $\alpha_{p,j}$ at $p$.

After step 2, we have $\mathcal{L},D_{S},f_{j}$ (still $D_{S}=0$ if $\\#S>1$)
with the property that, for any $p\in C^{\circ}$ and any $j\in[r+1]$, at most
one of the properties (i), (ii) hold.

  3. Step 3

($p\in C^{\circ}$ satisfies (i) for _more than one_ $j$)

Let $S_{1}$ be the set of $j\in[r+1]$ for which (i) holds. Let $\alpha_{j}$ be
the order to which $D_{\\{j\\}}$ contains $p$; in particular, $\alpha_{j}=0$
if $j\notin S_{1}$. Write $\alpha_{\max}$ for the largest of the $\alpha_{j}$,
and write $\alpha_{\text{tot}}$ for the sum of all of the $\alpha_{j}$.

As a section of $\mathcal{L}$, the order of vanishing of $f_{j}$ at $p$ is at
least $\alpha_{\text{tot}}-\alpha_{j}$, and, after step 2, we have equality
whenever $j\in S_{1}$. In particular, the common order of vanishing of the
$f_{j}$ is equal to $\alpha_{\text{tot}}-\alpha_{\max}$, which is strictly
positive exactly when $\\#S_{1}\geq 2$.

To remove the base-point at $p$, we wish therefore to twist down our sections
$f_{j}\in H^{0}(\mathcal{L})$ by $\alpha_{\text{tot}}-\alpha_{\max}$ at $p$.
After the twist, the new vanishing order of $f_{j}$ at $p$ (as a section of
$H^{0}(\mathcal{L})$) will be $\alpha^{\prime}_{j}:=\alpha_{\max}-\alpha_{j}$.
We will keep track of this new vanishing condition via the divisors $D_{S}$.

This is achieved more precisely as follows. First, write

$0=\alpha^{\prime}(1)<\cdots<\alpha^{\prime}(t)$

for the _distinct_ integers appearing among the $\alpha^{\prime}_{j}$. (Note
that $\alpha^{\prime}(t)$ is equal to $\alpha_{\max}$ unless $S_{1}=[r+1]$.)
Then, define the filtration

$[r+1]=S^{1}\supsetneq S^{2}\supsetneq\cdots\supsetneq S^{t}$

by

$S^{m}=\\{j\in[r+1]|\alpha^{\prime}_{j}\geq\alpha^{\prime}(m)\\}.$

Note that $S^{t}=[r+1]-S_{1}$ if $S_{1}\subsetneq[r+1]$.

Finally, we make the following modifications to our data.

     * •

$\mathcal{L}$ is replaced by
$\mathcal{L}(-(\alpha_{\text{tot}}-\alpha_{\max})p)$,

     * •

each $D_{\\{j\\}}$ is (temporarily) replaced by $D_{\\{j\\}}-\alpha_{j}p$,

     * •

for $m=2,3,\ldots,t$, the divisor $D_{[r+1]\backslash S^{m}}$ is replaced by
the divisor $D_{[r+1]\backslash
S^{m}}+(\alpha^{\prime}(m)-\alpha^{\prime}(m-1))p$, and

     * •

$f_{j}$ stays the same. (Note that it is a section of the same line bundle as
before; indeed, the multiplicity of $p$ in

$\sum_{S:j\notin S}D_{S}$

decreases by exactly $\alpha_{\text{tot}}-\alpha_{\max}$ in the previous two
modifications.)

After step 3, no $p\in C^{\circ}$ is a base-point. Therefore, the $f_{j}$
(viewed as sections of $\mathcal{L}$) define a map
$f^{\circ}:C^{\circ}\to\mathbb{P}^{r}$, with the property that the divisor

$D^{\prime}_{S}=\sum_{S^{\prime}\subseteq S}D_{S^{\prime}}$

(when restricted to $C^{\circ}$) is constrained to map to the torus-invariant
locus $\Lambda_{S}\subset\mathbb{P}^{r}$ defined earlier. There may be still
additional vanishing; for example, the $f_{j}$ may have unexpected vanishing
as sections of $\mathcal{L}\left(\sum_{j\notin S}D_{S}\right)$, but we will
not need to take this into account.

We now repeat the previous steps (with small modifications) on the points
$p_{i}$. The steps will be carried out for each $p_{i}$ independently; we fix
for the rest of this discussion an index $i$. We will furthermore assume for
simplicity as in the previous section that $x_{i}=[1:\cdots:1]$. Care must be
taken now to keep track of the effect of our twisting operations on the
conditions $V(x_{i})$.

  4. Step 4

($p_{i}$ satisfies (ii) for _all_ $j$)

This step is identical to step 1, with $p_{i}$ in place of $p$, and the
$D_{S}$ with $\\#S>1$ playing no role. After step 4, it will no longer be true
that $p_{i}$ satisfies (ii) for all $j$.

  5. Step 5

($p_{i}$ satisfies (i) for _all_ $j$)

This step is new. For a fixed $p_{i}$, $\alpha>0$ be the largest order to
which $p_{i}$ is contained in all of the $D_{j}$ simultaneously. Then, make
the following modifications:

     * •

$\mathcal{L}$ is replaced by $\mathcal{L}((-\alpha r)p)$,

     * •

each $D_{\\{j\\}}$ is replaced by $D_{\\{j\\}}-\alpha p$, while all $D_{S}$
with $\\#S>1$ remain unchanged, and

     * •

$f_{j}$ stays the same (note that it is a section of the same line bundle as
before).

After step 5, it will no longer be true that $p_{i}$ satisfies (i) for all
$j$.

######  Definition 42.

We say that $p_{i}$ is _inactive_ if either step 4 or 5 is run, that is, if
$p_{i}$ initially satisfies (i) for all $j$ or (ii) for all $j$, and that
$p_{i}$ is _active_ otherwise.

If $p_{i}$ is inactive, then the equations $V(x_{i})$ are automatically
satisfied. Therefore, after the twisting of either step 4 or 5, $V(x_{i})$
imposes no additional conditions on $f$.

######  Lemma 43.

Suppose that $p_{i}$ is active. Then, condition (i) holds for $j$ if and only
if (ii) does.

###### Proof.

For convenience of notation, we argue here in terms of the initial divisors
$D_{j}$, have not yet been modified in a neighborhood of $p_{i}$, and write
$\mathcal{L}^{\prime}=\mathcal{L}_{\text{init}}(-D_{1}-\cdots-D_{r+1})$.

Suppose that $p_{i}\in D_{j}$. We may also assume that there exists a
$j^{\prime}$ that $p_{i}\notin D_{j^{\prime}}$. Then,
$f_{j}(p_{i})=f_{j^{\prime}}(p_{i})=0$ in
$\mathcal{L}^{\prime}(D_{j}+D_{j^{\prime}})$, and because $p_{i}\notin
D_{j^{\prime}}$, we in fact have $f_{j}(p_{i})=0$ in
$\mathcal{L}^{\prime}(D_{j})$.

Conversely, if $f_{j}(p_{i})=0$ in $\mathcal{L}^{\prime}(D_{j})$, then, for
all $j^{\prime}\neq j$, we have $f_{j^{\prime}}(p_{i})=0$ in
$\mathcal{L}^{\prime}(D_{j}+D_{j^{\prime}})$. If $p_{i}\notin D_{j}$, then in
fact $f_{j^{\prime}}(p_{i})=0$ in $\mathcal{L}^{\prime}(D_{j^{\prime}})$, for
all $j^{\prime}$ (including $j$), contradicting the assumption that $p_{i}$ is
active. ∎

If $p_{i}$ is active, write $S^{i}_{\circ}:=\\{j\mid f_{j}\text{ satisfies
both (i) and (ii) at $p_{i}$}\\}$. Then, for any two $j,j^{\prime}\notin
S^{i}_{\circ}$, then $f_{j}(p_{i})=f_{j^{\prime}}(p_{i})\neq 0$ as sections of
$\mathcal{L}^{\prime}(D_{j}+D_{j^{\prime}})$.

######  Definition 44.

If $p_{i}$ is active, we say that $p_{i}$ is _regular_ if
$S^{i}_{\circ}=\emptyset$, that is, $p_{i}$ is not a base-point of $f$. We say
that $p_{i}$ is _wild_ otherwise.

The final two steps are only needed for wild and inactive $p_{i}$.

  6. Step 6

($p_{i}$ satisfies (i) and (ii) for _some_ $j$.)

Repeat step 2 with $p_{i}$ in place of $p$. The modifications are made exactly
for $j\in S^{i}_{\circ}$. Let $S^{i}_{\circ,1},S^{i}_{\circ,2}\subset
S^{i}_{\circ}$ denote the set of $j$ still satisfying conditions (i), (ii)
after step 6.

  7. Step 7

($p_{i}$ satisfies (i) for _more than one_ $j$)

Repeat step 3 with $p_{i}$ in place of $p$. Here, the subset $S^{i}_{\circ,1}$
plays the role of $S_{1}$ in step 3 above. Note that, by Lemma 43, step $7$ is
run only if step $6$ is.

After step 7, $f$ is finally base-point free everywhere, and therefore defines
a map $f:C\to\mathbb{P}^{r}$. We have already analyzed the incidence
conditions imposed on $C^{\circ}$ (after step 3); we now do so at the $p_{i}$.

First, suppose that $p_{i}$ is inactive. As we have noted, the condition
$V(x_{i})$ no longer imposes constraints on the $f_{j},D_{S}$ post-twisting.
On the other hand, after steps 6 and 7, the divisors $D_{S}$ containing
$p_{i}$ impose conditions on the intersection of $\text{im}(f)$ with the
torus-invariant boundary of $\mathbb{P}^{r}$ at $p_{i}$ in exactly the same
way as at the $p\in C^{\circ}$. Then, we summarize the situation:

For all non-empty $S\subsetneq[r+1]$, we have a divisor $D_{S}\subseteq C$ for
which

$f(D_{S})\subseteq\Lambda_{S}.$ (14)

(Note that $D_{S}\subseteq D_{S}^{\prime}$.) As we have already observed, we
may be forgetting various other constraints. For example, the $f_{j}$ may have
non-generic vanishing at various points of $C^{\circ}$ (including the
$D_{S}$), the $D_{S}$ may contain copies of the same points with various
multiplicites, and the $D_{S}$ may include the fixed inactive points $p_{i}$
in its support. The constraint (14) will turn out to be sufficient.

######  Lemma 45.

Suppose that $p_{i}$ is active. Then,

$f(p_{i})\in\langle x_{i},\Lambda_{S^{i}_{\circ}}\rangle=:\Lambda_{i},$ (15)

where $\langle-\rangle$ denotes linear span (so that $\Lambda_{i}$ has
dimension $\\#S^{i}_{\circ}$).

For regular $p_{i}$, note that (15) is simply the condition $f(p_{i})=x_{i}$.

###### Proof.

For any two $j,j^{\prime}\notin S^{i}_{\circ}$, then
$f_{j}(p_{i})=f_{j^{\prime}}(p_{i})\neq 0$ as sections of

$\mathcal{L}\left(-\sum_{S\notin j,j^{\prime}}D_{S}\right),$

and this property remains unchanged by steps 6 and 7 (and upon injection into
$\mathcal{L}$). ∎

We now come to the crux of the argument: combining the constraints (14) and
(15) on $f$ enumerated above, we will pass from $f:C\to\mathbb{P}^{r}$ to a
map $\bar{f}:C\to Y$, where $Y$ is the permutohedral variety of dimension $r$
defined in the previous section. Let $\bar{f}:C\to Y$ be unique lift of
$f:C\to\mathbb{P}^{r}$ to $Y$; this $\bar{f}$ exists because the sections
$f_{j}$ remain non-zero throughout the entire twisting process, so the image
of $f$ is not contained in any torus-invariant subvariety.

Write $\overline{\beta}:=\bar{f}_{*}[C]$, as well as
$\overline{d}=\deg(\mathcal{L})$ (after twisting down all base-points) and
$k_{S}=\deg(D_{S})$ for the degree of the divisor $D_{S}$. The following lemma
may be regarded as a numerical incarnation of (14).

######  Lemma 46.

We have

$\deg(\overline{\beta})\leq(r+1)\overline{d}\;-\sum_{\begin{subarray}{c}S\subset[r+1]\\\
1\leq\\#S\leq r-1\end{subarray}}(r-\\#S)k_{S},$

where the degree is measured against $K_{Y}^{\vee}$.

###### Proof.

Let $Y=Y_{r-1}\to Y_{r-2}\to\cdots\to Y_{0}=\mathbb{P}^{r}$ be the sequence of
blow-ups to obtain $Y$, where the blow-up $\rho_{s}:Y_{s+1}\to Y_{s}$ is at
the strict transforms of the $s$-dimensional torus-invariant subvarieties of
$\mathbb{P}^{r}$. Note that the map $f:C\to\mathbb{P}^{r}$ has degree
$\overline{d}$.

In the unique lift $f_{s}:C\to Y_{s}$ of $f$ to $Y_{s}$, for each $S$ with
$\\#S=s+1$, the divisor $D_{S}$ is constrained to map to strict transform of
$\Lambda_{S}$ in $Y_{s}$. Therefore, the blow-up $\rho_{s}$ decreases the
degree of $C$ by at least $(r-\\#S)k_{S}=(r-s-1)k_{S}$ for each such $S$, that
is,

$(f_{s})_{*}[C]\cdot K_{Y_{s}}^{\vee}-(f_{s+1})_{*}[C]\cdot
K_{Y_{s+1}}^{\vee}\geq(r-s-1)\sum_{\\#S=s+1}k_{S},$

from which the lemma follows. ∎

######  Lemma 47.

At active $p_{i}$, we have

$f(p_{i})\in\widetilde{\Lambda_{i}}.$ (16)

where $\widetilde{\Lambda_{i}}$ is the strict transform of $\Lambda_{i}$ in
$Y$.

Note that if $p_{i}$ is regular, then we may identify
$\widetilde{\Lambda_{i}}$ with $x_{i}$.

###### Proof.

As $x_{i}\in X$ is general, we may assume it does not lie in any torus-
invariant subvariety, and identify it both with a point of $Y$ and of
$\mathbb{P}^{r}$. Then, the subvariety $\widetilde{\Lambda_{i}}\subset Y$ may
be described in the language of §3.1.1 as the fiber containing $x_{i}$ of the
projection
$\rho_{(S_{\circ}^{i})^{c}}:Y\to\mathbb{P}(\mathbb{C}[(S_{\circ}^{i})^{c}])$.
That is, $\widetilde{\Lambda_{i}}$ is the set of points of $Y$ whose
coordinates corresponding to the complement of $S_{\circ}^{i}$ are equal to
those of $x_{i}$. When we take $x_{i}=[1:\cdots:1]$, this is to say that these
coordinates are all non-zero and equal to each other.

The claim now follows from the fact that, by construction, the sections
$f_{j}\in H^{0}(C,\mathcal{L})$ comprising $f:C\to\mathbb{P}^{r}$ have the
same order of vanishing for $j\in(S_{\circ}^{i})^{c}$. Therefore, twisting
$\mathcal{L}$ down by this order of vanishing yields a point $y_{k^{\prime}}$
in the sequence $y=(y_{0},\ldots,y_{k})$ for which the corresponding
coordinates are all non-zero and equal to each other, for the same reason as
in Lemma 45. ∎

If one or both of $S^{i}_{\circ,1},S^{i}_{\circ,2}$ are empty after step 6,
then in fact, $f(p_{i})$ is further constrained to lie in a proper subvariety
of $\widetilde{\Lambda_{i}}$; we will, however, not need this. Note in
particular that the expected number of conditions on $\bar{f}$ imposed by (16)
is $r-\\#S^{i}_{\circ}$.

######  Definition 48.

Let

$\tau^{\prime}:{\mathcal{M}}_{g,n}(Y,\bar{\beta})\to{\mathcal{M}}_{g,n}\times\prod_{p_{i}\text{
active}}\mathbb{P}(\mathbb{C}[(S^{i}_{\circ})^{c}])$

be the map obtained in the first factor by remembering the domain curve, and
in the second by evaluating at an active marked point $p_{i}$, and then
projecting via $\rho_{(S^{i}_{\circ})^{c}}$.

Note that the projection $\rho_{(S^{i}_{\circ})^{c}}$ is simply the blow-up
map $Y\to\mathbb{P}^{r}$ when $p_{i}$ is regular.

######  Lemma 49.

The expected relative dimension of $\tau^{\prime}$,

$\mathrm{vdim}({\mathcal{M}}_{g,n}(Y,\bar{\beta}))-(3g-3+n)-\sum_{p_{i}\
\text{active}}(r-\mathrm{\\#}S^{i}_{0}),$

is non-positive, and is zero only if our starting point
$f_{\text{init}}\in\mathbb{P}$ was base-point free, that is, in
$\mathbb{P}^{\circ}$.

###### Proof.

If $f_{\text{init}}\in\mathbb{P}$ to begin with, then

$\bar{\beta}\cdot K_{Y}^{\vee}\leq\beta\cdot K_{X}^{\vee},$

where $\beta$ is the original class
$d\mathsf{H}^{\vee}+\sum_{j}k_{j}\mathsf{E}_{j}^{\vee}\in H_{2}(X)$, with
equality whenever no two of the original sections $f_{j}\in
H^{0}(C,\mathcal{L}^{\prime}(D_{j}))$ share a common vanishing point. Thus,
the virtual dimension of ${\mathcal{M}}_{g,n}(Y,\bar{\beta})$ is at most the
dimension of ${\mathcal{M}}_{g,n}\times X^{n}$ by (1). In particular, the
quantity in question is non-positive.

In general, we will show that each of the steps of our twisting algorithm has
the effect of decreasing the quantity

$\Omega:=\left[(r+1)\deg(\mathcal{L})\;-\sum_{\begin{subarray}{c}S\subset[r+1]\\\
1\leq\\#S\leq
r-1\end{subarray}}(r-\\#S)\deg(D_{S})\right]-r(g-1)-\sum_{p_{i}\text{
active}}(r-\\#S^{i}_{\circ})$

The quantities $\deg(\mathcal{L}),\deg(D_{S})$ are taken to be those that are
changing throughout the twisting algorithm. As for the last term, we somewhat
abusively set the values of $(r-\\#S^{i}_{\circ})$ to be equal to $r$
initially (as if all marked points $p_{i}$ are initially regular), so that the
last summation is initially equal to $-rn$, and, by (1), $\Omega$ is initially
equal to zero. Then, if $p_{i}$ is inactive, the summand
$(r-\\#S^{i}_{\circ})$ is removed after either step 4 or 5 (whichever is
applied first). If $p_{i}$ is wild, then the value of $\\#S^{i}_{\circ}$ is
set to the correct one after step 6. If $p_{i}$ is regular, the summand
$(r-\\#S^{i}_{\circ})$ remains equal to $r$ throughout. The quantities
$\deg(\mathcal{L}),\deg(D_{S})$ are taken to be those that are changing
throughout the twisting algorithm.

By Lemma 46, the final value of $\Omega$ is an upper bound for the expected
relative dimension of $\tau^{\prime}$. Therefore, if
$f_{\text{init}}\notin\mathbb{P}^{\circ}$, that is, at least one step of the
algorithm is run, then the expected relative dimension of $\tau^{\prime}$ will
be strictly negative, as needed.

  1. Step 1

We decrease $\deg(\mathcal{L})$ by $\alpha>0$ and make no other changes, which
decreases $\Omega$ by $(r+1)\alpha$.

  2. Step 2

For each $j$, we decrease $\deg(\mathcal{L})$ and $\deg(D_{\\{j\\}})$ each by
$\alpha_{p,j}>0$, which decreases $\Omega$ by $2\alpha_{p,j}$.

  3. Step 3

First, decreasing $\deg(\mathcal{L})$ by $\alpha_{\text{tot}}-\alpha_{\max}$
decreases $\Omega$ by $(r+1)(\alpha_{\text{tot}}-\alpha_{\max})$, and
replacing each $D_{\\{j\\}}$ by $D_{\\{j\\}}-\alpha_{j}p$ increases it by
$(r-1)\alpha_{\text{tot}}$. Finally, the modification of $D_{[r+1]\backslash
S^{m}}$ decreases $\Omega$ by
$(\\#S^{m}-1)(\alpha^{\prime}(m)-\alpha^{\prime}(m-1))$. We thus need to show
that

$-2\alpha_{\text{tot}}+(r+1)\alpha_{\max}-\sum_{m=2}^{t}(\\#S^{m}-1)(\alpha^{\prime}(m)-\alpha^{\prime}(m-1))<0.$

Let

$\alpha(1)>\alpha(2)>\cdots>\alpha(t)$

denote the distinct integers among the $\alpha_{j}$; note that $\alpha(t)=0$
if and only if $S_{1}\subsetneq[r+1]$. Let $r_{1},\ldots,r_{t}$ denote the
number of times $\alpha(1),\ldots,\alpha(t)$ appear among the $\alpha_{j}$, so
that $r+1=r_{1}+\cdots+r_{t}$. We now have

$\displaystyle-2\alpha_{\text{tot}}+(r+1)\alpha_{\max}-\sum_{m=2}^{t}(\\#S^{m}-1)(\alpha^{\prime}(m)-\alpha^{\prime}(m-1))$
$\displaystyle=$
$\displaystyle-2\sum_{m=1}^{t}\alpha(m)r_{m}+(r+1)\alpha(1)-\sum_{m=2}^{t}(r_{m}+\cdots+r_{t}-1)(\alpha(m-1)-\alpha(m))$
$\displaystyle=$
$\displaystyle-\sum_{m=1}^{t}r_{m}\alpha(m)+\alpha(1)-\alpha(t)<0.$

  4. Step 4

As in step 1, we decrease $\deg(\mathcal{L})$ by $\alpha>0$ and make no other
changes, which decreases $\Omega$ by $(r+1)\alpha$. On the other hand, the
term $(r-\\#S^{i}_{\circ})$ goes away. Thus, the expected relative dimension
of $\tau^{\prime}$ goes down by $(r+1)\alpha-r>0$.

  5. Step 5

We decrease all of the $\deg(D_{\\{j\\}})$ by $\alpha>0$ and
$\deg(\mathcal{L})$ by $r\alpha$, decreasing $\Omega$ by $(r+1)\alpha$, and
possibly remove the term $(r-\\#S^{i}_{\circ})$ (if this was not done in the
previous step), so $\Omega$ goes down by at least $(r+1)\alpha-r>0$.

We examine the effect of the last two steps depending on the type of $p_{i}$.

  * •

if $p_{i}$ is inactive, then applications of steps 6 and 7 only decrease
$\Omega$ further, by steps 2 and 3.

  * •

if $p_{i}$ is regular, then no changes are made.

  * •

if $p_{i}$ is wild, then step 6 decreases $\deg(\mathcal{L})$ and
$\deg(D_{\\{j\\}})$ together by least 1 for each $j\in S^{i}_{\circ}$,
decreasing $\Omega$ by at least $2\cdot\\#S^{i}_{\circ}$. Furthermore, the
$-(r-\\#S^{i}_{\circ})$ is updated to the correct value (from $-r$ originally)
in this step. In total, the value of $\Omega$ goes down at least by

$2\cdot\\#S^{i}_{\circ}+(r-\\#S^{i}_{\circ})-r>0$

in step 6. In step 7, $\Omega$ can again only decrease, by step 3. (Note that
step $7$ does not affect the contribution of $-(r-\\#S^{i}_{\circ})$.)

∎

######  Lemma 50.

Every irreducible component of ${\mathcal{M}}_{g,n}(Y,\overline{\beta})$
dominating ${\mathcal{M}}_{g,n}\times\prod_{p_{i}\ \text{regular}}X$ (where we
may equivalently replace $X$ in the last factor with $Y$ or $\mathbb{P}^{r}$)
has expected dimension.

###### Proof.

We claim that there are at least $g+1$ regular points among the $p_{i}$.
Indeed, every point $p_{i}$ which is a base-point for our original
$f\in\mathbb{P}$ contributes at least 1 to the degree $d$ of our original line
bundle $\mathcal{L}$, but by assumption, we have $d\leq n-(g+1)$, so at least
$g+1$ regular points remain after our twisting algorithm.

Therefore, the conclusion follows from Lemma 1.

∎

###### Proof of Proposition 34.

Lemmas 49 and 50 at long last imply that $V(x_{1},\ldots,x_{n})$ is contained
in $\mathbb{P}^{\circ}$. Indeed, if this is not the case for some $f\in
V(x_{1},\ldots,x_{n})$, then applying the twisting algorithm to $f$ yields a
point on ${\mathcal{M}}_{g,n}(Y,\bar{\beta})$ moving in a family dominating
the target of $\tau^{\prime}$, but this is a contradiction due to dimension
reasons.

Similarly, for dimension reasons, we see that $f\in V(x_{1},\ldots,x_{n})$
must define a (base-point free) map $f:C\to X$ of curve class _exactly_
$\beta$, as unexpected vanishing of the $f_{j}$ can only decrease the degree
of $\beta$.

It remains to check that any $f\in V(x_{1},\ldots,x_{n})$ has no non-trivial
tangent vectors. It is elementary to check that such a tangent vector would
give rise to a non-trivial relative tangent vector of the map
$\tau:{\mathcal{M}}_{g,n}(X,\beta)\to{\mathcal{M}}_{g,n}\times X^{n}$ over a
general point, which contradicts Lemma 1. ∎

###  3.2 Integral formula on $S$

In this section, we push forward the formula of Theorem 10 to obtain Theorem
11. Up to setting $k_{i}=0$ for $i>\ell$, we can assume that $\ell=r+1$.

We start with a change of variables. Define

$\widetilde{\mathcal{E}}:=\mathcal{E}\otimes\mathcal{O}(N)=\bigoplus_{i=1}^{r+1}\pi_{*}(\mathcal{P}(-\mathcal{D}+N+\mathcal{D}_{i}))=\bigoplus_{i=1}^{r+1}\pi_{*}(\mathcal{P}(\Delta_{i})),$

where we have written

$N=\sum_{i=1}^{r+1}N_{i},\ \mathcal{D}=\sum_{i=1}^{r+1}\mathcal{D}_{i}\text{
and }\Delta_{i}=-\mathcal{D}+N+\mathcal{D}_{i}\text{ for }i=1,\ldots,r+1.$

Using $\widetilde{\mathcal{E}}$, we can rewrite the formula (6) as

$\mathsf{Tev}_{g,n,\beta}^{X}=\int_{\mathbb{P}(\widetilde{\mathcal{E}})}(\widetilde{\mathsf{H}}^{r}+\sigma_{1}\widetilde{\mathsf{H}}^{r-1}+\cdots.+\sigma_{r})^{n}$

where, by an abuse of notation,
$\widetilde{\mathsf{H}}=c_{1}(\mathcal{O}_{\mathbb{P}(\widetilde{\mathcal{E}})}(1))$.
Writing

$\displaystyle(\widetilde{\mathsf{H}}^{r}+\sigma_{1}\widetilde{\mathsf{H}}^{r-1}+\cdots.+\sigma_{r})^{n}$
$\displaystyle=\left(\frac{\prod_{i=1}^{r+1}(\widetilde{\mathsf{H}}+\eta_{i})-\sigma_{r+1}}{\widetilde{\mathsf{H}}}\right)^{n}$
$\displaystyle=\sum_{m=0}^{n}\binom{n}{m}(-1)^{m}\frac{\prod_{i=1}^{r+1}(\widetilde{\mathsf{H}}+\eta_{i})^{n-m}\sigma_{r+1}^{m}}{\widetilde{\mathsf{H}}^{n}}$
$\displaystyle=\sum_{m=0}^{n}\binom{n}{m}(-1)^{m}\frac{\prod_{i=1}^{r+1}(\sum_{a_{i}=0}^{n-m}\binom{n-m}{a_{i}}\widetilde{\mathsf{H}}^{n-m-
a_{i}}\eta_{i}^{a_{i}+m})}{\widetilde{\mathsf{H}}^{n}},$

we obtain

$\displaystyle\mathsf{Tev}_{g,n,\beta}^{X}$
$\displaystyle=\sum_{m=0}^{n}\binom{n}{m}(-1)^{m}\int_{\mathbb{P}(\widetilde{\mathcal{E}})}\frac{\prod_{i=1}^{r+1}(\sum_{a_{i}=0}^{n-m}\binom{n-m}{a_{i}}\widetilde{\mathsf{H}}^{n-m-
a_{i}}\eta_{i}^{a_{i}+m})}{\widetilde{\mathsf{H}}^{n}}$ (17)
$\displaystyle=\sum_{m=0}^{\min(n,k_{1},\ldots,k_{r+1})}\binom{n}{m}(-1)^{m}\int_{S}\left(\prod_{i=1}^{r+1}(1+\eta_{i})^{n-m}\eta_{i}^{m}\right)c(\widetilde{\mathcal{E}})^{-1},$

where in the last equality we have pushed forward to $S$. It remains to
compute $c(\widetilde{\mathcal{E}})$; this will be an application of
Grothendieck-Riemann-Roch.

####  3.2.1 Cohomology of
$\mathsf{Jac^{d}}(C)\times\mathsf{Sym}^{k_{1}}(C)\times\cdots\times\mathsf{Sym}^{k_{r+1}}(C)$

We now fix the notation necessary to compute the integral 17.

Let $e_{1},\ldots,e_{2g}$ be a symplectic basis of $H^{1}(C,\mathbb{Z})$, and
denote by $e_{1}^{\prime},\ldots,e_{2g}^{\prime}$ the corresponding classes in
$H^{1}(\mathsf{Jac^{d}}(C),\mathbb{Z})$ under the natural isomorphism

$H^{1}(\mathsf{Jac^{d}}(C),\mathbb{Z})\rightarrow H^{1}(C,\mathbb{Z}).$

Let also

$\Theta=\sum_{\alpha=1}^{g}e_{\alpha}^{\prime}e_{\alpha+g}^{\prime}$

be the class of the theta divisor on $\mathsf{Jac^{d}}(C)$ and set

$\gamma=-\left(\sum_{\alpha=1}^{g}e_{\alpha}^{\prime}e_{\alpha+g}-e_{\alpha+g}^{\prime}e_{\alpha}\right)\in
H^{1}(\mathsf{Jac^{d}}(C),\mathbb{Z})\otimes H^{1}(C,\mathbb{Z}).$

By [2, Chapter VIII], we have

$\mathrm{ch}(\mathcal{P})=1+d\mathsf{p}+\gamma-\Theta\mathsf{p}$ (18)

where $\mathsf{p}\in H^{2}(C,\mathbb{Z})$ is the point class.

Next, we recall MacDonald’s description [17] of the cohomology ring of
$\mathsf{Sym}^{k}(C)$. For $i=1,\ldots,r+1$, define classes
$\zeta_{i,1},\ldots,\zeta_{i,2g}\in H^{1}(\mathsf{Sym}^{k_{i}}(C),\mathbb{Z})$
for $\eta_{i}\in H^{2}(\mathsf{Sym}^{k_{i}}(C),\mathbb{Z})$ as the Künneth
components of the universal divisor $\mathcal{D}_{i}$:

$\mathcal{D}_{i}=\eta_{i}+\sum_{\alpha=1}^{g}(\zeta_{i,\alpha}e_{\alpha+g}-\zeta_{i,\alpha+g}e_{\alpha})+k_{i}\mathsf{p}.$

Note that $\eta_{i}=[N_{i}]$ as before. These generate the ring
$H^{*}(\mathsf{Sym}^{k_{i}}(C),\mathbb{Z})$, and setting
$\tau_{i,\alpha}=\zeta_{i,\alpha}\zeta_{i,\alpha+g}$, for every multiindex $I$
without repetitions, we have

$\int_{\mathsf{Sym}^{k_{i}}(C)}\eta_{i}^{k_{i}-|I|}\tau_{i,I}=1$

where $\tau_{i,I}=\prod_{\alpha\in I}\tau_{i,\alpha}$.

Set also

$\displaystyle\overline{y}_{i}=\sum_{\alpha=1}^{g}\sum_{\begin{subarray}{c}1\leq
j\leq r+1\\\ j\neq
i\end{subarray}}(\zeta_{j,\alpha}e_{\alpha+g}-\zeta_{j,\alpha+g}e_{\alpha}),$
$\displaystyle\overline{\tau}_{i}=\sum_{\alpha=1}^{g}\sum_{\begin{subarray}{c}1\leq
j_{1},j_{2}\leq r+1\\\ j_{1},j_{2}\neq
i\end{subarray}}\zeta_{j_{1},\alpha}\zeta_{j_{2},\alpha+g},$
$\displaystyle\overline{x}_{i}=\sum_{\alpha=1}^{g}\sum_{\begin{subarray}{c}1\leq
j\leq r+1\\\ j\neq
i\end{subarray}}e_{\alpha}^{\prime}\zeta_{j,\alpha+g}-e_{\alpha+g}^{\prime}\zeta_{j,\alpha},$
and $\displaystyle\overline{k}_{i}=\sum_{\begin{subarray}{c}1\leq j\leq r+1\\\
j\neq i\end{subarray}}k_{j},$

and write

$\Delta_{i}=\eta_{i}-\overline{y}_{i}-\overline{k}_{i}.$

The following lemma will be used in Section §3.2.2.

######  Lemma 51.

The following identities hold.

  1. (i)

$\Delta_{i}^{h}=\eta_{i}^{h}-\eta_{i}^{h-1}(h\overline{y}_{i}+h\overline{k}_{i}\mathsf{p})+\eta_{i}^{h-2}(-h(h-1)\overline{\tau}_{i}\mathsf{p})$

  2. (ii)

$\gamma\overline{y}_{i}=\overline{x}_{i}\mathsf{p}$

###### Proof.

Identity (i) is proved by induction on $h$ using
$\overline{y}_{i}^{2}=-2\overline{\tau}_{i}\mathsf{p}$. Identity (ii) is
straightforward. ∎

####  3.2.2 Computation of $c(\widetilde{\mathcal{E}})$

We now express $c(\widetilde{\mathcal{E}})$ in terms of the classes defined
above. Fix an index $i$ with $1\leq i\leq r+1$.

######  Proposition 52.

We have

$\mathrm{ch}(\pi_{*}(\mathcal{P}(\Delta_{i}))=\mathrm{exp}(\eta_{i})(1-g+d-\overline{k}_{i}-\overline{\tau}_{i}-\Theta+\overline{x}_{i}).$

###### Proof.

By Grothendieck-Riemann-Roch and (18), we have

$\displaystyle\mathrm{ch}(\pi_{*}(\mathcal{P}(\Delta_{i}))$
$\displaystyle=\pi_{*}(\mathrm{ch}(\mathcal{P}(\Delta_{i}))\cdot\mathsf{Td}_{C})$
$\displaystyle=\pi_{*}((1+d\mathsf{p}+\gamma-\mathsf{p}\Theta)\cdot\mathrm{exp}(\Delta_{i})\cdot(1+(1-g)\mathsf{p}).$

We compute

$\displaystyle\pi_{*}(\mathrm{exp}(\Delta_{i}))$
$\displaystyle=\pi_{*}\left(\sum_{h=0}^{\infty}\frac{\Delta_{i}^{h}}{h!}\right)$
$\displaystyle=\pi_{*}\left(\sum_{h=0}^{\infty}\frac{-h\overline{k}_{i}}{h!}\eta_{i}^{h-1}\mathsf{p}-\frac{h(h-1)}{h!}\eta_{i}^{h-2}\overline{\tau}_{i}\mathsf{p}\right)$
$\displaystyle=\mathrm{exp}(\eta_{i})(-\overline{k}_{i}-\overline{\tau}_{i})$

where in the second equality we used identity (i) in Lemma 51. Similarly, we
have

$\displaystyle\pi_{*}((d-\Theta)\mathsf{p}\
\mathrm{exp}(\Delta_{i}))=\mathrm{exp}(\eta_{i})(d-\Theta),$
$\displaystyle\pi_{*}(-\gamma\
\mathrm{exp}(\Delta_{i}))=\mathrm{exp}(\eta_{i})\overline{x}_{i},$
$\displaystyle\pi_{*}((1-g)\mathsf{p}\
\mathrm{exp}(\Delta_{i})\mathrm{ch}(\mathcal{P}))=(1-g)\mathrm{exp}(\eta_{i}).$

Here, in the second equality we have used identity (ii) in Lemma 51. After
summing all the obtained contributions, we obtain the claim. ∎

######  Corollary 53.

$c(\widetilde{\mathcal{E}})=\prod_{i=1}^{r+1}(1+\eta_{i})^{1-g+d-\overline{k}_{i}}\cdot\mathrm{exp}\left(\frac{-\overline{\tau}_{i}-\Theta+\overline{x}_{i}}{1+\eta_{i}}\right)$

###### Proof.

This follows immediately from the formula

$c(V)=\sum_{h=1}^{\infty}(-1)^{h-1}(h-1)!\cdot\mathrm{ch}_{h}(V)$

valid for any vector bundle $V$ on any scheme. ∎

Substituting 53 into (17) completes the proof of Theorem 11, namely that

$\mathsf{Tev}_{g,n,\beta}^{X}=\sum_{m=0}^{\min(n,k_{1},\ldots,k_{r+1})}\binom{n}{m}(-1)^{m}\int_{S}\prod_{i=1}^{r+1}(1+\eta_{i})^{n-m-1+g-d+\overline{k}_{i}}\eta_{i}^{m}\cdot\mathrm{exp}\left(\frac{\overline{\tau}_{i}+\Theta-\overline{x}_{i}}{1+\eta_{i}}\right).$

###  3.3 Specialization to genus 0

We now specialize Theorem 11 to $g=0$ to prove Theorem 12.

We continue to assume that $\ell=r+1$. Identify

$\mathsf{Sym}^{k_{i}}(\mathbb{P}^{1})=\mathbb{P}^{k_{i}}\text{ for
}i=1,\ldots,r+1.$

Then, the class $\eta_{i}$ is the first Chern class of
$\mathcal{O}_{\mathbb{P}^{k_{i}}}(1)$ and Theorem 11 gives:

$\displaystyle\mathsf{Tev}_{0,n,\beta}^{X}$
$\displaystyle=\sum_{m=0}^{\min(n,k_{1},\ldots,k_{r+1})}\binom{n}{m}(-1)^{m}\int_{\prod_{i=1}^{r}\mathbb{P}^{k_{i}}}\prod_{i=1}^{r+1}(1+\eta_{i})^{n-m-1-d+\overline{k}_{i}}\eta_{i}^{m}$
$\displaystyle=\sum_{m=0}^{\min(n,k_{1},\ldots,k_{r+1})}\binom{n}{m}(-1)^{m}\prod_{i=1}^{r+1}\mathrm{Coeff}\left((1+\eta_{i})^{n-m-1-d+\overline{k}_{i}};\eta_{i}^{k_{i}-m}\right)$
$\displaystyle=\sum_{m=0}^{\min(n,k_{1},\ldots,k_{r+1})}\binom{n}{m}(-1)^{m}\prod_{i=1}^{r+1}\binom{n-m-d-1+\overline{k}_{i}}{k_{i}-m}.$

This concludes the proof of Theorem 12.

###  3.4 Specialization to $X={\mathsf{Bl}}_{q}(\mathbb{P}^{r})$

Theorem 11 also admits a reasonably elegant specialization to
$X={\mathsf{Bl}}_{q}(\mathbb{P}^{r})$. We take now $k_{2}=\ldots=k_{r+1}=0$
and write $k=k_{1}$, $\eta=\eta_{1}$, and $\mathsf{E}=\mathsf{E}_{1}$, so that
$\tau=\overline{\tau}_{2}=\cdots=\overline{\tau}_{r+1}$ and
$x=\overline{x}_{2}=\cdots=\overline{x}_{r+1}$.

We have

$\displaystyle\mathsf{Tev}_{g,n,d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}}^{X}$
$\displaystyle=\int_{\mathsf{Jac^{d}}(C)\times\mathsf{Sym}^{k}(C)}(1+\eta)^{n-1+g-d}\cdot\mathrm{exp}\left(\frac{\Theta}{1+\eta}\right)\cdot\mathrm{exp}(\tau+\Theta-x)$
$\displaystyle=\int_{\mathsf{Jac^{d}}(C)\times\mathsf{Sym}^{k}(C)}A(\eta)\cdot\mathrm{exp}(\tau
B(\eta)+\Theta C(\eta)+xD(\eta)),$

where

$\begin{cases}A(\eta)&=(1+\eta)^{n-1+g-d},\\\ B(\eta)&=-r,\\\
C(\eta)&=\frac{r\eta+(r+1)}{1+\eta},\\\ D(\eta)&=r.\end{cases}$

######  Lemma 54.

Let $A(\eta),B(\eta),C(\eta)$ and $D(\eta)$ be polynomials in $\eta$. Then,

$\displaystyle\int_{\mathsf{Jac^{d}}(C)\times\mathsf{Sym}^{k}(C)}A(\eta)\cdot\mathrm{exp}(\tau
B(\eta)+$ $\displaystyle\Theta C(\eta)+xD(\eta))$
$\displaystyle=\mathrm{Coeff}\left(A(\eta)(C(\eta)(1+\eta B(\eta)-\eta
D(\eta)^{2})^{g};\eta^{k}\right)$

###### Proof.

Notice that for $i+j=g$, we have

$\Theta^{i}x^{2j}=\sum_{\begin{subarray}{c}I=(i_{1},\ldots,i_{j})\\\ 1\leq
i_{1}<\ldots<i_{j}\leq g\end{subarray}}i!(-1)^{j}(2j)!e_{1}^{\prime}\cdots
e_{2g}^{\prime}\tau_{I},$

where $\tau_{I}=\prod_{i\in I}\tau_{i}$ and
$\tau_{\alpha}=\zeta_{1,\alpha}\zeta_{1,\alpha+g}$ for $\alpha=1,\ldots,g$.
Therefore, we can expand

$\displaystyle\int_{\mathsf{Jac^{d}}(C)\times\mathsf{Sym}^{k}(C)}A(\eta)\cdot\mathrm{exp}(\tau
B(\eta)+\Theta C(\eta)+xD(\eta))$
$\displaystyle=\sum_{m=0}^{g}(-1)^{g-m}\sum_{\begin{subarray}{c}I=(i_{1},\ldots,i_{g-m})\\\
1\leq i_{1}<\cdots<i_{g-m}\leq
g\end{subarray}}\int_{\mathsf{Sym}^{k}(C)}A(\eta)C(\eta)^{m}D(\eta)^{2(g-m)}\cdot\mathrm{exp}(\tau
B(\eta))\tau_{I}.$

Call

$\widetilde{A}(\eta)=A(\eta)C(\eta)^{m}D(\eta)^{2(g-m)}.$ (19)

We have

$\displaystyle\int_{\mathsf{Sym}^{k}(C)}\widetilde{A}(\eta)\cdot\mathrm{exp}(\tau
B(\eta))\tau_{I}$
$\displaystyle=\sum_{h=0}^{\infty}\int_{\mathsf{Sym}^{k}(C)}\widetilde{A}(\eta)\frac{\tau_{I}\tau^{h}B(\eta)^{h}}{h!}$
$\displaystyle=\sum_{h=0}^{\infty}\int_{\mathsf{Sym}^{k}(C)}\widetilde{A}(\eta)\left(\sum_{\begin{subarray}{c}J:|J|=h\\\
J\cap I=\emptyset\end{subarray}}\tau_{I}\tau_{J}\right)B(\eta)^{h}$
$\displaystyle=\sum_{h=0}^{g-|I|}\binom{g-|I|}{|J|}\cdot\mathrm{Res}_{\eta=0}\left\\{\frac{\widetilde{A}(\eta)B(\eta)^{h}}{\eta^{k-h-|I|+1}}d\eta\right\\}$
$\displaystyle=\mathrm{Res}_{\eta=0}\left\\{\frac{\widetilde{A}(\eta)}{\eta^{k-|I|+1}}(1+\eta
B(\eta))^{g-|I|}d\eta\right\\}.$

The claim follows then by substituting (19) and using the combinatorial
identity

$\sum_{m=0}^{g}\binom{g}{m}(C(\eta)(1+\eta
B(\eta))^{m}(-D(\eta)^{2}\eta)^{g-m}=\left(C(\eta)(1+\eta B(\eta))-\eta
D(\eta)^{2}\right)^{g}$

∎

From Lemma 54, we obtain the residue formula

$\displaystyle\mathsf{Tev}_{g,n,d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}}^{X}$
$\displaystyle=\mathrm{Coeff}\left((1+\eta)^{n-1-d}(2r\eta+(r+1))^{g};\eta^{k}\right),$

and the right hand side can be expanded as

$\displaystyle\sum_{m=0}^{m}\binom{g}{m}(2r)^{m}(r+1)^{g-m}\binom{n-1-d}{k-m}$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{m}\binom{g}{m}(2r)^{m}\sum_{h=0}^{g-m}\binom{g-m}{h}(1-r)^{g-m-h}(2r)^{h}\binom{n-1-d}{k-m}$
$\displaystyle=$
$\displaystyle\sum_{a=0}^{g}\binom{g}{a}(2r)^{a}(1-r)^{g-a}\sum_{m=0}^{a}\binom{a}{m}\binom{n-1-d}{k-m}$
$\displaystyle=$
$\displaystyle\sum_{a=0}^{g}\binom{g}{a}(2r)^{a}(1-r)^{g-a}\binom{g}{a}\binom{n-1-d+a}{k},$

where in the second equality we have set $a=m+h$ and in the third equality we
have used applied Vandermonde identity

${\displaystyle{m+n\choose r}=\sum_{k=0}^{r}{m\choose k}{n\choose r-k}}\text{
for all }r,n,m\in\mathbb{Z}_{\geq 0}.$

Finally, replacing $a$ with $g-m$, we obtain Theorem 13.

##  4 Virtual counts for $X={\mathsf{Bl}}_{q}(\mathbb{P}^{r})$

In this section, we establish the virtual count of Theorem 14.

###  4.1 Preliminaries on $QH^{*}(X)$

The computation of the invariants ${\mathsf{vTev}}^{X}_{g,n,\beta}$ is reduced
to a calculation in the quantum cohomology ring $QH^{*}(X,\mathbb{Q})$ in [4].
See [11] for an introduction. We now take
$X={\mathsf{Bl}}_{q}(\mathbb{P}^{r})$.

Because $X$ is Fano, by [18, Proposition 2.2], we have

$QH^{*}(X,\mathbb{Q})\cong\mathbb{Q}[\mathsf{C}]\otimes_{\mathbb{Q}}H^{*}(X,\mathbb{Q})$

as $\mathbb{Q}[\mathsf{C}]$-modules. Here, $\mathsf{C}\subseteq
H_{2}(X,\mathbb{Z})$ is the cone of effective curves in $X$, and by [8,
Proposition 4.1], we have

$\mathsf{C}=\mathbb{Z}_{\geq
0}(\mathsf{H}^{\vee}+\mathsf{E}^{\vee})\oplus\mathbb{Z}_{\geq
0}(-\mathsf{E}^{\vee}).$

We will denote by $\star$ the product in $QH^{*}(X,\mathbb{Q})$.

Let $\Sigma\subseteq\mathbb{R}^{r}$ be toric fan of $X$. Its generators are
$v_{j}=e_{j+1}$ for $j=0,1,\ldots,r-1$, along with
$v_{r}=-(e_{1}+\cdots+e_{r})$ and $v=-v_{r}$.

The sets

$S=\\{v_{0},\ldots,v_{r-1}\\}\text{ and }T=\\{v,v_{r-1}\\}$

are the only two primitive sets in the sense of [13, Definition 1.1] and,
using [13, Theorem 1.2], from these two sets we respectively obtain the
following two relations in $QH^{*}(X,\mathbb{Q})$:

$\displaystyle(\mathsf{H}-\mathsf{E})^{\star r}$
$\displaystyle=q^{-\mathsf{E}^{\vee}}\mathsf{E};$ (20)
$\displaystyle\mathsf{H}\star\mathsf{E}$
$\displaystyle=q^{\mathsf{H}^{\vee}+\mathsf{E}^{\vee}}.$

######  Lemma 55.

The set

$1,(\mathsf{H}-\mathsf{E}),\ldots,(\mathsf{H}-\mathsf{E})^{\star
r-1},\mathsf{E}\star(\mathsf{H}-\mathsf{E}),\ldots.,\mathsf{E}\star(\mathsf{H}-\mathsf{E})^{r-1}$

is a basis of $QH^{*}(X,\mathbb{Q})$ as $\mathbb{Q}[\mathsf{C}]$-module.

###### Proof.

From the second relation of (20), we have

$\mathsf{E}^{\star
2}=-(\mathsf{H}-\mathsf{E})\star\mathsf{E}+q^{\mathsf{H}^{\vee}+\mathsf{E}^{\vee}},$

so the above collection spans $QH^{*}(X,\mathbb{Q})$. The fact that they are
linearly independent follows from [13, Theorem 1.2]. ∎

Consider the homogeneous basis

$\mathsf{H}^{r},\mathsf{H}^{r-1},\ldots,1,\mathsf{E}^{r-1},\ldots,\mathsf{E}$
(21)

of $H^{*}(X,\mathbb{Q})$. We now express each of these classes in
$QH^{*}(X,\mathbb{Q})$ as linear combinations of the basis elements of Lemma
55 using [13, Theorem 1.6]. In order for [13, Theorem 1.6] to apply, we need
the following result:

######  Lemma 56.

Every toric subvariety of $X$ is Fano. Furthermore, for every map $\pi:X\to
X^{\prime}$, where $X^{\prime}$ is toric and $\pi$ is the blowuup of an
irreducible toric subvariety, we have that $X^{\prime}$ is Fano.

###### Proof.

Immediate consequence of [13, Theorem 3.9]. ∎

######  Lemma 57.

We have

$\displaystyle\mathsf{H}^{i}$
$\displaystyle=\mathsf{H}\star(\mathsf{H}-\mathsf{E})^{\star(i-1)}\text{ for
}i=1,\ldots,r,\text{ and}$ $\displaystyle\mathsf{E}^{i}$
$\displaystyle=(-1)^{i-1}\mathsf{E}\star(\mathsf{H}-\mathsf{E})^{\star(i-1)}\text{
for }i=1,\ldots,r-1$

in $QH^{*}(X,\mathbb{Q})$.

###### Proof.

We start with $\mathsf{H}^{i}$. Let $\alpha_{i}$ be the cone generated by
$v_{r},\ldots,v_{i+1}$ for $i=1,\ldots,r$. We have that $\alpha_{i}$
corresponds to a toric subvariety of class $\mathsf{H}^{r-i}$. Then, [13,
Theorem 1.6] and the fact the only exceptional set special for $\alpha_{i}$ is
the empty set yields the desired equality.

The proof for $\mathsf{E}^{i}$ is similar. This time, $\alpha_{i}$ is the cone
generated by $v$ and $v_{r-2},\ldots,v_{r-i}$ for $i=1,\ldots,r-1$ (when
$i=1$, we just have $\alpha_{1}=\langle v\rangle$), so $\alpha_{i}$
corresponds to $(-1)^{i+1}\mathsf{E}^{i}$. ∎

###  4.2 The quantum Euler class of $X$

By definition, the quantum Euler class of $X$ is the image of the Künneth
decomposition of the diagonal in $X\times X$ under the natural product map

$H^{*}(X,\mathbb{Q})\otimes
H^{*}(X,\mathbb{Q})\xrightarrow{\star}QH^{*}(X,\mathbb{Q}).$

This is a canonically defined element of $QH^{*}(X,\mathbb{Q})$, first
introduced by Abrams in [1] (see also [4, 5]).

The computation of the virtual invariants
${\mathsf{vTev}}^{X}_{g,n,d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}}$ is related
to $\Delta$ in [4, Theorem 1.3] by the formula:

${\mathsf{vTev}}^{X}_{g,n,d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}}=\mathrm{Coeff}(\mathsf{P}^{\star
n}\star\Delta^{\star g},\mathsf{P}q^{d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}}).$

where $\mathsf{P}$ is the point class in $X$.

######  Theorem 58.

We have

$\Delta=(2r)\mathsf{P}-(r-1)q^{-\mathsf{E}^{\vee}}\mathsf{E}.$

in $QH^{*}(X,\mathbb{Q})$.

###### Proof.

Let

$1,\mathsf{H},\ldots,\mathsf{H}^{r},(-1)^{r-1}\mathsf{E},\ldots.,(-1)^{r-1}\mathsf{E}^{r-1}.$

be the dual basis to 21 w.r.t. the Poincarè pairing. Then

$\displaystyle\Delta=$ $\displaystyle\mathsf{P}\star
1+\cdots+\mathsf{H}^{r-i}\star\mathsf{H}^{i}+\cdots+1\star\mathsf{P}+(-1)^{r-1}(\mathsf{E}\star\mathsf{E}^{r-1}+\cdots+\mathsf{E}^{r-1}\star\mathsf{E})$
$\displaystyle=$ $\displaystyle 2\mathsf{P}+(r-1)\mathsf{H}^{\star
2}\star(\mathsf{H}-\mathsf{E})^{\star(r-2)}-(r-1)\mathsf{E}^{\star
2}\star(\mathsf{H}-\mathsf{E})^{\star(r-2)}$ $\displaystyle=$ $\displaystyle
2\mathsf{P}+(r-1)\left((\mathsf{H}-\mathsf{E})^{r}+\mathsf{E}^{2}\star(\mathsf{H}-\mathsf{E})^{r-2}+2\mathsf{E}\star(\mathsf{H}-\mathsf{E})^{\star(r-1)}\right)$
$\displaystyle-(r-1)\mathsf{E}^{\star
2}\star(\mathsf{H}-\mathsf{E})^{\star(r-2)}$ $\displaystyle=$
$\displaystyle(2r)\mathsf{P}-(r-1)q^{-\mathsf{E}^{\vee}}\mathsf{E},$

where in the second equality we have used Lemma 57, in the third equality we
have written $\mathsf{H}=\mathsf{H}-\mathsf{E}+\mathsf{E}$, and in the forth
equality we have used the first relation in 20 and the fact that, by Lemma 57,
we have

$\mathsf{P}=\mathsf{H}\star(\mathsf{H}-\mathsf{E})^{\star(r-1)}=(\mathsf{H}-\mathsf{E})^{\star
r}+\mathsf{E}\star(\mathsf{H}-\mathsf{E})^{\star(r-1)}.$

This concludes the proof. ∎

###  4.3 Proof of Theorem 14

Finally, we can compute

$\displaystyle{\mathsf{vTev}}^{X}_{g,n,d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}}$
$\displaystyle=\mathrm{Coeff}(\mathsf{P}^{\star n}\star\Delta^{\star
g},\mathsf{P}q^{d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}})$
$\displaystyle=\sum_{m=0}^{g}(2r)^{g-m}(1-r)^{m}\binom{g}{m}\mathrm{Coeff}(\mathsf{P}^{\star
n+g-m}\star\mathsf{E}^{\star
m},\mathsf{P}q^{d\mathsf{H}^{\vee}+(k+m)\mathsf{E}^{\vee}}),$

and the proof of Theorem 14 is reduced to the following.

######  Lemma 59.

For $m,k\in\mathbb{Z}_{\geq 0}$, $\ell,d\in\mathbb{Z}_{>0}$ with $\ell-d-m>0$,
$d\geq k$ and satisfying

$(r+1)d-(r-1)k=r(\ell-1),$ (22)

we have

$\mathrm{Coeff}(\mathsf{P}^{\star\ell-m}\star\mathsf{E}^{\star
m},\mathsf{P}q^{d\mathsf{H}^{\vee}+(k+m)\mathsf{E}^{\vee}})=\binom{\ell-
d-m-1}{k}.$

######  Remark 60.

Note that

$\binom{\ell-d-m-1}{k}=0$

unless $\ell-d-m-1\geq k$, which, using (22), is equivalent to $d\geq(2r-1)k$.
In particular, we get $0$ unless $\ell\geq r+2$.

###### Proof of Lemma 59 .

We proceed by induction on $m$. Suppose that $m=0$. Then, conditions (1), (3),
and (5) are all satisfied with $n=\ell$, so

$\mathrm{Coeff}(\mathsf{P}^{\star\ell},\mathsf{P}q^{d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}})={\mathsf{vTev}}^{X}_{0,\ell,d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}}=\binom{\ell-d-1}{k}$

by Theorems 12 and 23.

Suppose that $m=1$. If $\ell>r+1$, then we can write

$\displaystyle\mathsf{P}^{\star(\ell-1)}\star\mathsf{E}$
$\displaystyle=\mathsf{P}^{\star(\ell-2)}\star(\mathsf{H}-\mathsf{E})^{\star(r-1)}q^{\mathsf{H}^{\vee}+\mathsf{E}^{\vee}}$
$\displaystyle=\mathsf{P}^{\star(\ell-3)}\star\mathsf{H}\star(\mathsf{E}q^{-\mathsf{E}^{\vee}})\star(\mathsf{H}-\mathsf{E})^{\star(r-2)}q^{\mathsf{H}^{\vee}+\mathsf{E}^{\vee}}$
$\displaystyle=\mathsf{P}^{\star(\ell-3)}\star(\mathsf{H}-\mathsf{E})^{\star(r-2)}q^{2\mathsf{H}^{\vee}+\mathsf{E}^{\vee}}$
$\displaystyle\ \vdots$
$\displaystyle=\mathsf{P}^{\star(\ell-r-1)}q^{r\mathsf{H}^{\vee}+\mathsf{E}^{\vee}}.$

Taking the coefficient of
$\mathsf{P}q^{d\mathsf{H}^{\vee}+(k+1)\mathsf{E}^{\vee}}$ on both sides and
applying the $m=0$ case gives the claim. If instead $\ell<r+2$, then the same
chain of equalities yields

$\mathsf{P}^{\star(\ell-1)}\star\mathsf{E}=(\mathsf{H}-\mathsf{E})^{\star(r+1-\ell)}q^{(\ell-1)\mathsf{H}^{\vee}+\mathsf{E}^{\vee}}.$

In particular, the coefficient of
$\mathsf{P}q^{d\mathsf{H}^{\vee}+(k+1)\mathsf{E}^{\vee}}$ is $0$. The claimed
equality then follows from Remark 60.

Finally, suppose that $m\geq 2$. Write

$\displaystyle\mathsf{P}^{\star(\ell-m)}\mathsf{E}^{\star m}$
$\displaystyle=\mathsf{P}^{\star(\ell-m-1)}\mathsf{E}^{\star(m-1)}(\mathsf{H}-\mathsf{E})^{\star(r-1)}q^{\mathsf{H}^{\vee}+\mathsf{E}^{\vee}}$
$\displaystyle=\mathsf{P}^{\star(\ell-m)}\mathsf{E}^{\star(m-2)}q^{\mathsf{H}^{\vee}+\mathsf{E}^{\vee}}-\mathsf{P}^{\star(\ell-m-1)}\mathsf{E}^{\star(m-1)}q^{\mathsf{H}^{\vee}}.$

Taking the coefficient of
$\mathsf{P}q^{d\mathsf{H}^{\vee}+(k+1)\mathsf{E}^{\vee}}$ on both sides and
using the inductive hypothesis completes the proof. ∎

This concludes the proof of Theorem 14.

######  Remark 61.

The condition $n-d\geq 1$ in Theorem 14 (i.e., the condition $\ell-m-d\geq 1$
in Lemma 59 ) is necessary. For example, when $r=2$ and $n=g=d=k=1$, we have

$\displaystyle{\mathsf{vTev}}^{X}_{g,n,d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}}$
$\displaystyle=\mathrm{Coeff}(\mathsf{P}\star\Delta,q^{d\mathsf{H}^{\vee}+k\mathsf{E}^{\vee}}\mathsf{P})$
$\displaystyle=4\cdot\mathrm{Coeff}(q^{\mathsf{H}^{\vee}}\mathsf{H},q^{\mathsf{H}^{\vee}+\mathsf{E}^{\vee}}\mathsf{P})-\mathrm{Coeff}(q^{\mathsf{H}^{\vee}+\mathsf{E}^{\vee}}(\mathsf{H}-\mathsf{E}),q^{\mathsf{H}^{\vee}+2\mathsf{E}^{\vee}}\mathsf{P})$
$\displaystyle=0$

while the right-hand side of Theorem 14 is equal to $1$.

######  Remark 62.

When $k=0$, in Lemma 59 we have

$d=\frac{r(\ell-1)}{r+1}\in\mathbb{Z},$

and so it must be that $\ell>r+1$. In particular, as one would expect, the
same proof gives

$\mathsf{vTev}^{X}_{g,n,d\mathsf{H}^{\vee}}=(r+1)^{g}=\mathsf{vTev}^{\mathbb{P}^{r}}_{g,n,d\mathsf{H}^{\vee}}$

for all $g\geq 0$ and $n,d>1$ satisfying (1).

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