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I'm reading the Atanas Atanasov's course notes of Joe Harris' course Geometry of Algebraic Curves and have a question about a suggested modification of an dimension counting argument applying methods from deformation theory.

On page 22 one consideres a version of Hurwitz scheme

$$ V_{d,g}:= \{(X, f: X \to \mathbb{P}^2) \ \vert \ X \text{ curve of genus } g, f \text{ has degree } d \text{ and is birational } \\ \text{ onto a plane curve with } \delta \text{ nodes } \} $$

together with two canonical canonical projection maps $V_{d,g} \to M_g $ (to the 'naive' moduli set) and $V_{d,g} \to \mathbb{P}^{\delta} \backslash \Delta$.

Rather elementary considerations in the script show that $\dim V_{d,g}=3+g−1$ if $d(d+3)/2 \ge 3 \delta$ but the Remark 4.2 says:

There is a serious problem with this argument if $3 \delta> d(d + 3)/2 $ but this can be fixed using deformation theory.

Namely, the counting method in the script used as intermediate equalities $\dim V_{d,g}=d(d+3)/2−3\Delta+2\Delta=3+g−1$. Of course for $3 \delta> d(d + 3)/2 $ these considerations make no any sense, but the final equality between left and right is known to be still true.

Does somebody know how to fix this gap using deformation theoretic arguments as suggested in the remark 4.2?

I'm reading the Atanas Atanasov's course notes of Joe Harris' course Geometry of Algebraic Curves and have a question about a suggested modification of an dimension countinging argument applying methods from deformation theory.

On page 22 one consideres a version of Hurwitz scheme

$$ V_{d,g}:= \{(X, f: X \to \mathbb{P}^2) \ \vert \ X \text{ curve of genus } g, f \text{ has degree } d \text{ and is birational } \\ \text{ onto a plane curve with } \delta \text{ nodes } \} $$

together with two canonical canonical projection maps $V_{d,g} \to M_g $ (to the 'naive' moduli set) and $V_{d,g} \to \mathbb{P}^{\delta} \backslash \Delta$.

Rather elementary considerations in the script show that $\dim V_{d,g}=3+g−1$ if $d(d+3)/2 \ge 3 \delta$ but the Remark 4.2 says:

There is a serious problem with this argument if $3 \delta> d(d + 3)/2 $ but this can be fixed using deformation theory.

Namely, the counting method in the script used as intermediate equalities $\dim V_{d,g}=d(d+3)/2−3\Delta+2\Delta=3+g−1$. Of course for $3 \delta> d(d + 3)/2 $ these considerations make no any sense, but the final equality between left and right is known to be still true.

How to fix this gap using deformation theoretic arguments as suggested in the remark 4.2 in detail?

I'm reading the Atanas Atanasov's course notes of Joe Harris' course Geometry of Algebraic Curves and have a question about a suggested modification of an dimension counting argument applying methods from deformation theory.

On page 22 one consideres a version of Hurwitz scheme

$$ V_{d,g}:= \{(X, f: X \to \mathbb{P}^2) \ \vert \ X \text{ curve of genus } g, f \text{ has degree } d \text{ and is birational } \\ \text{ onto a plane curve with } \delta \text{ nodes } \} $$

together with two canonical canonical projection maps $V_{d,g} \to M_g $ (to the 'naive' moduli set) and $V_{d,g} \to \mathbb{P}^{\delta} \backslash \Delta$.

Rather elementary considerations in the script show that $\dim V_{d,g}= 3d+g-1$ if $d(d+3)/2 \ge 3 \delta$ but the Remark 4.2 says:

There is a serious problem with this argument if $3 \delta> d(d + 3)/2 $ but this can be fixed using deformation theory.

Does somebody know how to fix this gap using deformation theoretic arguments as suggested in the remark 4.2?

I'm reading the Atanas Atanasov's course notes of Joe Harris' course Geometry of Algebraic Curves and have a question about a suggested modification of an dimension counting argument applying methods from deformation theory.

On page 22 one consideres a version of Hurwitz scheme

$$ V_{d,g}:= \{(X, f: X \to \mathbb{P}^2) \ \vert \ X \text{ curve of genus } g, f \text{ has degree } d \text{ and is birational } \\ \text{ onto a plane curve with } \delta \text{ nodes } \} $$

together with two canonical canonical projection maps $V_{d,g} \to M_g $ (to the 'naive' moduli set) and $V_{d,g} \to \mathbb{P}^{\delta} \backslash \Delta$.

Rather elementary considerations in the script show that $\dim V_{d,g}=3+g−1$ if $d(d+3)/2 \ge 3 \delta$ but the Remark 4.2 says:

There is a serious problem with this argument if $3 \delta> d(d + 3)/2 $ but this can be fixed using deformation theory.

Namely, the counting method in the script used as intermediate equalities $\dim V_{d,g}=d(d+3)/2−3\Delta+2\Delta=3+g−1$. Of course for $3 \delta> d(d + 3)/2 $ these considerations make no any sense, but the final equality between left and right is known to be still true.

Does somebody know how to fix this gap using deformation theoretic arguments as suggested in the remark 4.2?

I'm reading the Atanas Atanasov's course notes of Joe Harris' course Geometry of Algebraic Curves and have a question about a suggested modification of an dimension counting argument applying methods from deformation theory.

On page 22 one consideres a version of Hurwitz scheme

$$ V_{d,g}:= \{(X, f: X \to \mathbb{P}^2) \ \vert \ X \text{ curve of genus } g, f \text{ has degree } d \text{ and is birational } \\ \text{ onto a plane curve with } \delta \text{ nodes } \} $$

together with two canonical canonical projection maps $V_{d,g} \to M_g $ (to the 'naive' moduli set) and $V_{d,g} \to \mathbb{P}^{\delta} \backslash \Delta$.

Rather elementary considerations in the script show that $\dim V_{d,g}= 3d+g-1$ if $d(d+3)/2 \ge 3 \delta$ but the Remark 4.2 says:

There is a serious problem with this argument if $3 \delta> d(d + 3)/2 $ but this can be fixed using deformation theory.

Could somebody elaborate this deformation theoretic argument fixing the gap the remark 4.2 is refering to?

I'm reading the Atanas Atanasov's course notes of Joe Harris' course Geometry of Algebraic Curves and have a question about a suggested modification of an dimension counting argument applying methods from deformation theory.

On page 22 one consideres a version of Hurwitz scheme

$$ V_{d,g}:= \{(X, f: X \to \mathbb{P}^2) \ \vert \ X \text{ curve of genus } g, f \text{ has degree } d \text{ and is birational } \\ \text{ onto a plane curve with } \delta \text{ nodes } \} $$

together with two canonical canonical projection maps $V_{d,g} \to M_g $ (to the 'naive' moduli set) and $V_{d,g} \to \mathbb{P}^{\delta} \backslash \Delta$.

Rather elementary considerations in the script show that $\dim V_{d,g}= 3d+g-1$ if $d(d+3)/2 \ge 3 \delta$ but the Remark 4.2 says:

There is a serious problem with this argument if $3 \delta> d(d + 3)/2 $ but this can be fixed using deformation theory.

Does somebody know how to fix this gap using deformation theoretic arguments as suggested in the remark 4.2?