## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 71, Number 1 (2019), 299-308.

### The combinatorics of Lehn's conjecture

Alina MARIAN, Dragos OPREA, and Rahul PANDHARIPANDE

#### Abstract

Let $S$ be a nonsingular projective surface equipped with a line bundle $H$. Lehn's conjecture is a formula for the top Segre class of the tautological bundle associated to $H$ on the Hilbert scheme of points of $S$. Voisin has recently reduced Lehn's conjecture to the vanishing of certain coefficients of special power series. The first result here is a proof of the vanishings required by Voisin by residue calculations (A. Szenes and M. Vergne have independently found the same proof). Our second result is an elementary solution of the parallel question for the top Segre class on the symmetric power of a nonsingular projective curve $C$ associated to a higher rank vector bundle $V$ on $C$. Finally, we propose a complete conjecture for the top Segre class on the Hilbert scheme of points of $S$ associated to a higher rank vector bundle on $S$ in the $K$-trivial case.

#### Note

The first author was supported by the NSF through grant DMS 1601605. The second author was supported by the NSF through grant DMS 1150675. The third author was supported by the Swiss National Science Foundation and the European Research Council through grants SNF-200020-162928, ERC-2012-AdG-320368-MCSK and ERC-2017-AdG-786580-MACI. The third author was also supported by SwissMAP and the Einstein Stiftung in Berlin.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 71, Number 1 (2019), 299-308.

**Dates**

Received: 3 September 2017

First available in Project Euclid: 4 October 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1538640045

**Digital Object Identifier**

doi:10.2969/jmsj/78747874

**Mathematical Reviews number (MathSciNet)**

MR3909922

**Zentralblatt MATH identifier**

07056565

**Subjects**

Primary: 14C05: Parametrization (Chow and Hilbert schemes)

Secondary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] 14Q05: Curves 14Q10: Surfaces, hypersurfaces

**Keywords**

Lehn conjecture Hilbert scheme of points

#### Citation

MARIAN, Alina; OPREA, Dragos; PANDHARIPANDE, Rahul. The combinatorics of Lehn's conjecture. J. Math. Soc. Japan 71 (2019), no. 1, 299--308. doi:10.2969/jmsj/78747874. https://projecteuclid.org/euclid.jmsj/1538640045