Affiliation:
1. Department of Mathematics , 1810 Tufts University , 177 College Avenue , Medford , MA 02155 , USA
Abstract
Abstract
We consider the problem of enumerating maps 𝑓 of degree 𝑑 from a fixed general curve 𝐶 of genus 𝑔 to
P
r
\mathbb{P}^{r}
satisfying incidence conditions of the form
f
(
p
i
)
∈
X
i
f(p_{i})\in X_{i}
, where
p
i
∈
C
p_{i}\in C
are general points and
X
i
⊂
P
r
X_{i}\subset\mathbb{P}^{r}
are general linear spaces.
We give a complete answer in the case where the
X
i
X_{i}
are points, where the counts are known as the “Tevelev degrees” of
P
r
\mathbb{P}^{r}
.
These were previously known only when
r
=
1
r=1
, when 𝑑 is large compared to
r
,
g
r,g
, or virtually in Gromov–Witten theory.
We also give a complete answer in the case
r
=
2
r=2
with arbitrary incidence conditions.
Our main approach studies the behavior of complete collineations under various degenerations.
Funder
National Science Foundation
Reference29 articles.
1. R. Beheshti, B. Lehmann, C. Lian, E. Riedl, J. Starr and S. Tanimoto,
On the asymptotic enumerativity property for virtual Tevelev degrees of Fano manifolds, preprint (2023), https://arxiv.org/abs/2310.15252; to appear in Forum Math. Sigma.
2. A. Berget and A. Fink,
Equivariant Chow classes of matrix orbit closures,
Transform. Groups 22 (2017), no. 3, 631–643.
3. A. Bertram,
Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian,
Internat. J. Math. 5 (1994), no. 6, 811–825.
4. A. Bertram, G. Daskalopoulos and R. Wentworth,
Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians,
J. Amer. Math. Soc. 9 (1996), no. 2, 529–571.
5. A. Buch and R. Pandharipande,
Tevelev degrees in Gromov–Witten theory, preprint (2021), https://arxiv.org/abs/2112.14824.