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Consider $J^{g-1}$, the variety of degree $g-1$ line bundles on a compact Riemann surface of genus $g$. Recall that $J^{g-1}$ is a torsor for the Jacobian, thus has dimension $g$. We can produce elements in $J^{g-1}$ by choosing $g-1$ points $p_1,\ldots,p_{g-1}$ and constructing the associated line bundle to the divisor $p_1+\cdots+p_{g-1}$. In this way, we obtain a $g-1$-dimensional family of elements of $J^{g-1}$, forming the so-called $\Theta$ divisor. Elements of the $\Theta$ divisor may be distinguished from other points in $J^{g-1}$ by the fact that they represent line bundles admitting sections, i.e. $h^0(L)\neq 0$.

For degree $g-1$ line bundles, the Riemann-Roch theorem gives $h^0(L)=h^1(L)$, so over $J^{g-1}$ we see that $h^0(L),h^1(L)$ are generically zero and jump along the $\Theta$ divisor. One might imagine that there are vector bundles $V,W$ over $J^{g-1}$ of the same rank, representing with fibres over $L\in J^{g-1}$ given by Dolbeault $0$- and $1$-forms with coefficients in $L$ respectively, together with a bundle map $\overline{\partial}:V\rightarrow W$ which is generically an isomorphism but drops rank along $\Theta$; then $\det \overline{\partial}$ would be a section of a line bundle $\det V^*\otimes\det W$ over $J^{g-1}$ which cuts out $\Theta$. This line bundle turns out to be a well-defined object called the determinant line bundle and it was introduced by Quillen.

Quillen's construction of the line bundle proceeds by replacing the family of (2-term) Dolbeault complexes parametrized by $J^{g-1}$ by a quasi-isomorphic family of finite-dimensional (2-term) complexes and taking the determinant line bundle of this. The finite-dimensional replacement for the Dolbeault complex is given by the Dolbeault operator acting on forms lying in the first few eigenspaces of the Laplacian, roughly speaking. Then as we move along $J^{g-1}$, some of the eigenvalues may hit zero and we have a jump of $h^0(L)$.

For a good list of references for the above, see determinant line bundle (nlab).

My question: What is a simple, modern construction of the determinant line bundle on $J^{g-1}$, perhaps one which uses the tools of algebraic geometry? It may well be tautological from some point of view, in which case I would probably be unsatisfied. Also, if you wish to rhapsodize on determinant line bundles, please feel free.

My motivation: Many of the classic papers by Beauville, Narasimhan, Ramanan etc. on moduli of vector bundles expend a lot of effort working with determinant line bundles and extracting information from them in quite ad-hoc and clever ways. I'd like to understand these better.

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# Construction of the determinant line bundle on the degree $g-1$ Picard variety

Consider $J^{g-1}$, the variety of degree $g-1$ line bundles on a compact Riemann surface of genus $g$. Recall that $J^{g-1}$ is a torsor for the Jacobian, thus has dimension $g$. We can produce elements in $J^{g-1}$ by choosing $g-1$ points $p_1,\ldots,p_{g-1}$ and constructing the associated line bundle to the divisor $p_1+\cdots+p_{g-1}$. In this way, we obtain a $g-1$-dimensional family of elements of $J^{g-1}$, forming the so-called $\Theta$ divisor. Elements of the $\Theta$ divisor may be distinguished from other points in $J^{g-1}$ by the fact that they represent line bundles admitting sections, i.e. $h^0(L)\neq 0$.

For degree $g-1$ line bundles, the Riemann-Roch theorem gives $h^0(L)=h^1(L)$, so over $J^{g-1}$ we see that $h^0(L),h^1(L)$ are generically zero and jump along the $\Theta$ divisor. One might imagine that there are vector bundles $V,W$ over $J^{g-1}$ of the same rank, representing Dolbeault $0$- and $1$-forms respectively, together with a bundle map $\overline{\partial}:V\rightarrow W$ which is generically an isomorphism but drops rank along $\Theta$; then $\det \overline{\partial}$ would be a section of a line bundle $\det V^*\otimes\det W$ over $J^{g-1}$ which cuts out $\Theta$. This line bundle turns out to be a well-defined object called the determinant line bundle and it was introduced by Quillen.

Quillen's construction of the line bundle proceeds by replacing the family of (2-term) Dolbeault complexes parametrized by $J^{g-1}$ by a quasi-isomorphic family of finite-dimensional (2-term) complexes and taking the determinant line bundle of this. The finite-dimensional replacement for the Dolbeault complex is given by the Dolbeault operator acting on forms lying in the first few eigenspaces of the Laplacian, roughly speaking. Then as we move along $J^{g-1}$, some of the eigenvalues may hit zero and we have a jump of $h^0(L)$.

For a good list of references for the above, see determinant line bundle (nlab).

My question: What is a simple, modern construction of the determinant line bundle on $J^{g-1}$, perhaps one which uses the tools of algebraic geometry? It may well be tautological from some point of view, in which case I would probably be unsatisfied. Also, if you wish to rhapsodize on determinant line bundles, please feel free.

My motivation: Many of the classic papers by Beauville, Narasimhan, Ramanan etc. on moduli of vector bundles expend a lot of effort working with determinant line bundles and extracting information from them in quite ad-hoc and clever ways. I'd like to understand these better.