\[ A ( S ) \ne S \, ; \qquad \qquad \qquad \qquad \qquad \qquad ( 2 . 1 4 ) \]

since \( A \neq { \mathrm { I d } } \), such a subspace exists. Consider the vector bundle V on Y constructed in (2.11) using S. As before, P(V ) denotes the projective bundle over Y parametrizing the lines in the fibers of V .

There is an embedding

\[ \delta : P ( V ) \longrightarrow Q \qquad \qquad \qquad \qquad \qquad ( 2 . 1 5 ) \]

which we will now describe. For the map f in (2.2), the image \( f \circ \delta ( P ( V ) ) \) is the point given by the quotient

\[ 0 \, \longrightarrow \, { \widehat V } \, \longrightarrow \, { \mathcal O } _ { X } ^ { \oplus r } \, \longrightarrow \, \bigl ( \bigoplus _ { i = 1 } ^ { d _ { p } - 1 } ( { \mathcal O } _ { X } ^ { \oplus r } ) _ { x _ { i } } / H _ { i } \bigr ) \oplus \bigl ( { \mathcal O } _ { X } ^ { \oplus r } \bigr ) _ { x _ { 0 } } / S \bigr ) \, \longrightarrow \, 0 \, . \]

where \( H _ { i } \) are the hyperplanes fixed above; in particular, \( f \circ \delta \) is a constant map. Note that bV is an extension of the vector bundle V to X. For any point \( y \in Y \) and any point in the fiber \( y ^ { \prime } \in P ( V ) _ { y } \), consider the short exact sequence on X

\[ 0 \, \longrightarrow \, E \, \longrightarrow \, \widehat { V } ^ { * } \, \longrightarrow \, ( \oplus _ { j = 1 } ^ { d _ { z } - 1 } ( \widehat { V } _ { y _ { j } } ^ { * } / L _ { j } ^ { \perp } ) \oplus ( \widehat { V } _ { y } ^ { * } / L ( y ^ { \prime } ) ^ { \perp } ) \, \longrightarrow \, 0 \, , \]

where \( L ( y ^ { \prime } ) \, \subset \, V _ { y } \) is the line in \( V _ { y } \) corresponding to the above point \( y ^ { \prime } \), and \( L _ { j } ^ { \perp } \, \subset \, \widehat { V } _ { y _ { j } } ^ { * } \) (respectively, \( L ( y ^ { \prime } ) ^ { \perp } \subset \widehat { V } _ { y } ^ { * } ) \) is the annihilator of \( L _ { j } \) (respectively, \( L ( y ^ { \prime } ) ) \); note that \( \widehat { V } _ { y _ { j } } \) is identified with \( \mathbb { C } ^ { r } \) and \( \widehat { V } _ { y } \) is identified with \( V _ { y } \). Therefore, we have

\[ \widehat { V } \hookrightarrow E ^ { * } . \]

The map \( \delta \) in (2.15) sends any \( y ^ { \prime } \) to the above extension \( E ^ { * } \) of bV constructed from \( y ^ { \prime } \).

From (2.10) we know that \( \mathrm { P G L } ( r , \mathbb { C } ) \) is contained in \( \operatorname { A u t } ( p ^ { - 1 } ( U ) ) \) with

\[ 0 \oplus { \mathfrak { g } } \, \subset \, { \mathfrak { g } } \oplus { \mathfrak { g } } \, = \, H ^ { 0 } ( p ^ { - 1 } ( U ) , \, T ( p ^ { - 1 } ( U ) ) ) \]

as its Lie algebra. This action of \( \mathrm { P G L } ( r , \mathbb { C } ) \) on \( p ^ { - 1 } ( U ) \) clearly preserves the intersection \( \delta ( P ( V ) ) \cap p ^ { - 1 } ( U ) \). Therefore, if the action of the element A in (2.13) extends to Q, then the extended action must preserve the image \( \delta ( P ( V ) ) \).

On the other hand, from Lemma 2.3 we know that the action of A on \( P ( V ) | _ { Y \backslash \{ y _ { 0 } \} } \) does not extend to P(V ) because (2.14) holds. This completes the proof of the theorem.

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# 3. Holomorphic maps from a symmetric product

Proposition 3.1. Let X and Y be compact connected Riemann surface with

\[ \operatorname { g e n u s } ( X ) \, \geq \, \operatorname { g e n u s } ( Y ) \, \geq \, 2 \, . \]

If there is a nonconstant holomorphic map \( \beta \; : \; \mathrm { S y m } ^ { d } ( Y ) \; \longrightarrow \; X \), then \( d \)= 1, and \( \beta \) is an isomorphism.

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.

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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