Proof. Let \( \beta \, : \, \mathrm { S y m } ^ { d } ( Y ) \, \longrightarrow \, X \) be a nonconstant holomorphic map. Let

\[ \beta ^ { * } \, : \, H ^ { 0 } ( X , \, \Omega _ { X } ^ { 1 } ) \, \longrightarrow \, H ^ { 0 } ( \mathrm { S y m } ^ { d } ( Y ) , \, \Omega _ { \mathrm { S y m } ^ { d } ( Y ) } ^ { 1 } ) \]

be the pull-back of 1–forms defined by \( \omega \longmapsto \beta ^ { * } \omega \). This homomorphism \( \beta ^ { * } \) is injective, because \( \beta \) is surjective. Since

\[ \mathrm { d i m } \, H ^ { 0 } ( \mathrm { S y m } ^ { d } ( Y ) , \, \Omega _ { \mathrm { S y m } ^ { d } ( Y ) } ^ { 1 } ) \, = \, \mathrm { g e n u s } ( Y ) \]

[Ma, p. 322, (4.3)], the injectivity of \( \beta ^ { * } \) implies that \( \operatorname { g e n u s } ( Y ) \; \geq \; \operatorname { g e n u s } ( X ) \). Therefore, the given condition \( \operatorname { g e n u s } ( X ) \ \geq \operatorname { g e n u s } ( Y ) \) implies that

• \( \operatorname { g e n u s } ( X ) \, = \, \operatorname { g e n u s } ( Y ) \), and

• the above homomorphism \( \beta ^ { * } \) is an isomorphism.

If \( d \geq \) 2, the wedge product

\[ \wedge ^ { 2 } H ^ { 0 } ( \mathrm { S y m } ^ { d } ( Y ) , \, \Omega _ { \mathrm { S y m } ^ { d } ( Y ) } ^ { 1 } ) \, \longrightarrow \, H ^ { 0 } ( \mathrm { S y m } ^ { d } ( Y ) , \, \Omega _ { \mathrm { S y m } ^ { d } ( Y ) } ^ { 2 } ) \]

is a nonzero homomorphism [Ma, p. 325, (6.3)]. On the other hand, the wedge product on \( H ^ { 0 } ( X , \, \Omega _ { X } ^ { 1 } ) \) is the zero homomorphism because \( H ^ { 0 } ( X , \, \Omega _ { X } ^ { 2 } ) \; = \; 0 \). In other words, \( \beta ^ { * } \) is not compatible with the wedge product operation on holomorphic 1-forms if \( d \geq \) 2. So we conclude that \( d \) = 1.

Since \( \mathrm { g e n u s } ( X ) \, = \, \mathrm { g e n u s } ( Y ) \), from Riemann–Hurwitz formula for Euler characteristic if follows that \( \mathrm { d e g r e e } ( \beta ) \) = 1. In other words, \( \beta \) is an isomorphism.



Let \( X ^ { \prime } \) be a compact connected Riemann surface of genus at least two. Fix positive integers \( r ^ { \prime } \ge \) 2, \( d _ { p } ^ { \prime } \) and \( d _ { z } ^ { \prime } \). Let

\[ \boldsymbol { \mathcal { Q } } ^ { \prime } \, = \, \boldsymbol { \mathcal { Q } } _ { X } ^ { \prime } ( \boldsymbol { r } ^ { \prime } , d _ { p } ^ { \prime } . d _ { z } ^ { \prime } ) \]

be the corresponding generalized quot scheme (see (2.2)).

Proposition 3.2. If the two varieties \( \mathcal { Q } ^ { \prime } \) and Q (constructed in (2.2)) are isomorphic, then X is isomorphic to \( X ^ { \prime } \).

Proof. Assume that \( Q \) 0 and Q are isomorphic. We will show that X and \( X ^ { \prime } \) are isomorphic.

Let \( \eta \, : \, \mathrm { S y m } ^ { d _ { p } } ( X ) \times \mathrm { S y m } ^ { d _ { z } } ( X ) \, \longrightarrow \, \mathrm { P i c } ^ { d _ { p } } ( X ) \times \mathrm { P i c } ^ { d _ { z } } ( X ) \) be the morphism defined by

\[ \left( ( x _ { 1 } \, , \, \cdot \, \cdot \, , x _ { d _ { p } } ) \, , ( y _ { 1 } \, , \, \cdot \, \cdot \, , y _ { d _ { z } } ) \right) \longmapsto \left( { \mathcal { O } } _ { X } ( x _ { 1 } + \, . \, . \, . + x _ { d _ { p } } ) \, , { \mathcal { O } } _ { X } ( y _ { 1 } + \, . \, . \, . \, + y _ { d _ { z } } ) \right) . \]

Since the general fiber of the map p in (2.5) is a product of copies of projective spaces, the composition

\[ \eta \circ p : \, \mathcal { Q } \longrightarrow \operatorname { P i c } ^ { d _ { p } } ( X ) \times \operatorname { P i c } ^ { d _ { z } } ( X ) \]

is the Albanese map for Q, as there is no nonconstant holomorphic map from a projec-tive space to an abelian variety. In particular, the Albanese variety of Q is of dimension