Second main theorem for holomorphic curves on annuli with arbitrary families of hypersurfaces

1.   Introduction

The main result in Nevanlinna theory is called the second fundamental theorem. In 1925, Nevanlinna ^[1] established the second main theorem for meromorphic functions on the complex plane $\mathbb{C}$. In 1933, H. Cartan ^[2] proved the second main theorem for holomorphic curves with targets in the form of hyperplanes in the general position in the complex projective spaces $\mathbf{P}^{n}(\mathbb{C})$. In 1983, E. I. Nochka ^[3] proved the second main theorem in the case of hyperplanes in the $N$-subgeneral position in $\mathbf{P}^{n}(\mathbb{C})$ with ramification. In 2004, M. Ru ^[4] established the second main theorem for holomorphic curves with targets in the form of hypersurfaces in the general position in $\mathbf{P}^{n}(\mathbb{C})$ without ramification. In 2009, Ru ^[5] made further extension to algebraically nondegenerate holomorphic curves into an arbitrary smooth complex projective variety. Since that time, the problem of investigation of the characteristics of holomorphic maps has attracted the attention of numerous authors.

In this paper, we mainly consider the case for holomorphic curves from the doubly connected domain into $\mathbf{P}^{n}(\mathbb{C})$. By the doubly connected mapping theorem ^[6], each doubly connected domain in $\mathbb{C}$ is conformally equivalent to the annulus $\mathbb{A}(r, R) = \{z:\; r < |z| < R\}, \; 0\leq r < R\leq +\infty$. We need only consider two cases: $r = 0, R = +\infty$ simultaneously and $0 < r < R < +\infty$. In the latter case the homothety $z\mapsto \frac{z}{\sqrt{r R}}$ reduces the given domain to the annulus $\left\{z:\; \frac{1}{R_{0}} < |z| < R_{0} \right\}$ with $R_{0} = \sqrt{\frac{R}{r}}$. Thus, in the two cases every annulus is invariant with respect to the inversion $z\mapsto \frac{1}{z}$. Observing the above facts, Khrystiyanyn and Kondratyuk ^[7,8] indicated the way to extend the value distribution of Nevanlinna theory to meromorphic functions in annuli.

Let $R_{0} < +\infty$ be a fixed positive real number or $+\infty$ and let

be an annuli in $\mathbf{C}$. Moreover, for any real number r such that $1 < r < R_{0}$, we denote

and

Let $f = (f_{0}:\ldots:f_{n+1})$ be a holomorphic curve from the annuli $\mathbb{A}$ into the complex projective space $\mathbf{P}^{n}(\mathbb{C})$. For $1 < r < R_{0}$, the characteristic function of $f$ is defined by

where

Remark 1. The above definition is independent, up to an additive constant, of the choice of the reduced representation of $f$.

Let $D$ be a hypersurface in $\mathbf{P}^{n}(\mathbb{C})$ of degree $d$. Let $Q$ be the homogeneous polynomial of degree $d$, defining $D$. The proximity function of $f$ is defined by

To be within an additive constant, this definition is independent of the choice of the reduced representation of $f$ and the choice of the defining polynomial $Q$.

Further, for $j = 1, 2$, by $n_{j, f}(r, D)$ we denote the number of zeros of $Q \circ f$ in $\mathbb{A}_{j, r}$, counting multiplicity, and let $n_{j, f}^{M}(r, D)$ be the number of zeros of $Q \circ f$ in the disk $\mathbb{A}_{j, r}$, where any zero of multiplicity greater than $M$ is “truncated” and counted as if it only had multiplicity $M$. We set

The integrated counting and truncated counting functions are defined by

When we want to emphasize $Q$, we sometimes also write $N_{f}(r, D)$ as $N_{f}(r, Q)$ and $N_{f}^{[M]}(r, D)$ as $N_{f}^{[M]}(r, Q)$.

In the present paper, we set the small error term by

In 2015, H. T. Phuong and N. V. Thin ^[9] considered the extension of the second main theorem for holomorphic curves from $\mathbb{A}$ into $\mathbf{P}^{n}(\mathbb{C})$ crossing a finite set of fixed hyperplanes in general position.

Theorem A. (^[9]) Let $f: \mathbb{A} \to \mathbf{P}^{n}(\mathbb{C})$ be a linearly nondegenerate holomorphic curve. Let $H_{1}, \ldots, H_{q}$ be hyperplanes in $\mathbf{P}^{n}(\mathbb{C})$, located in general position, then we have

Here and in the following, the notation “$||\; \mathcal{P}$” means that if $R_{0} = +\infty$, then the assertion $\mathcal{P}$ holds for all $r\in (1, +\infty)$ outside a set $\mathbb{A}_{r}'$ with $\int_{\mathbb{A}_{r}'}r^{\lambda -1} dr < +\infty$. At the same time, if $R_{0} < +\infty$, then the assertion $\mathcal{P}$ holds for all $r\in (1, R_{0})$ outside a set $\mathbb{A}_{r}'$ with $\int_{\mathbb{A}_{r}'}\frac{1}{(R_{0}-r)^{\lambda+1}}dr < +\infty$, where $\lambda \geq 0$.

Recently, J.L. Chen and T.B. Cao ^[10] obtained the second main theorem for holomorphic curves on annuli crossing a finite set of moving hyperplanes in sub-general position in $\mathbf{P}^{n}(\mathbb{C})$. It is well known that all known second main theorems and uniqueness results hold under the conditions that the hyperplanes or hypersurfaces are located in general position (or in sub-general position). More recently, S.D. Quang ^[11] introduced the notion of “distributive constant” $\Delta_{V}$ of a family of moving hypersurfaces with respect to a subvariety $V$ of $\mathbf{P}^{n}(\mathbb{C})$ and generalized some results to the case of meromorphic mappings into a projective subvariety $V$ and an arbitrary family of moving hypersurfaces. Motivated by this new notion, we show the second main theorem for holomorphic curves from the annulus into the complex projective space which is ramified over an arbitrary family of hypersurfaces. We also give an explicit estimate for the level of truncation.

For the purpose of this article, we recall some definitions. For a subvariety $V$ and an analytic subset $S$ of $\mathbf{P}^{n}(\mathbb{C})$, the codimension of $S$ in $V$ is defined by

According to Quang ^[11], we give the following definition.

Definition 1. Let $\left\{D_{j}\right\}_{j = 1}^{q}$ be the hypersurfaces in $\mathbf{P}^{n}(\mathbb{C})$. Denote by $Q$ the index set $\{1, \ldots, q\}$. Let $V$ be a subvariety of $\mathbf{P}^{n}(\mathbb{C})$ of dimension $k$. Assume that $V$ is not contained in any $D_{j}\; (j\in Q)$. We define the distributive constant of $\left\{D_{j}\right\}_{j = 1}^{q}$ with respect to $V$ by

Definition 2. Let $\left\{D_{j}\right\}_{j = 1}^{q}$ be the hypersurfaces in $\mathbf{P}^{n}(\mathbb{C})$. Denote by $Q$ the index set $\{1, \ldots, q\}$. Let $N \geq n$ and $q \geq N+1$. The family $\left\{D_{j}\right\}_{j = 1}^{q}$ is said to be in $N$-subgeneral position with respect to $V$ if for every subset $R \subset Q$ with the cardinality $\sharp R = N+1$, then

If $N = \dim V$ then we say that $\left\{D_{j}\right\}_{j = 1}^{q}$ is in general position with respect to $V$.

Remark 2. If the family $\left\{D_{j}\right\}_{j = 1}^{q}$ is in $N$-subgeneral position with respect to $V$, then $\Delta_{V}\leq N- \dim V + 1$ (see ^[11]).

In this paper, we establish the following second main theorem for holomorphic curves from the annulus into a complex projective variety intersecting an arbitrary family of hypersurfaces. The proof of our result follows from the paper ^[12,13].

Theorem 1. Let $V$ be a projective subvariety of $\mathbf{P}^{n}(\mathbb{C})$ of dimension $k$. Let $f: \mathbb{A}\to V$ be an algebraically nondegenerate holomorphic curve with $0 < R_{0}\leq + \infty.$ Let $\{D_{j}\}_{j = 1}^{q}$ be $q$ hypersurfaces in $\mathbf{P}^{n}(\mathbb{C})$ with $\deg D_{j} = d_{j}\; \; (1 \leq j \leq q)$. Let $d$ be the least common multiple of $d_{j}'s$, i.e., $d = \operatorname{lcm}(d_{ 1}, \ldots, d_{ q})$. Let $\Delta_{V}$ be the distributive constant of $\{D_{j}\}_{j = 1}^{q}$ with respect to $V$, then for any $\varepsilon > 0$,

where

with $l = (k+1)q!.$

Here and in the following, $\left \lfloor x\right \rfloor$ denotes the greatest integer less than or equal to the real number $x.$

By Remark 2, we have an immediate corollary.

Corollary 1. Let $V$ be a projective subvariety of $\mathbf{P}^{n}(\mathbb{C})$ of dimension $k$. Let $f: \mathbb{A}\to V$ be an algebraically nondegenerate holomorphic curve with $0 < R_{0}\leq + \infty$. Let $D_{1}, \ldots, D_{q}$ be the hypersurfaces in $\mathbf{P}^{n}(\mathbb{C})$, located in $N$-subgeneral position with respect to $V$ with $d_{j}: = \operatorname{deg} D_{j}(1 \leq j \leq q)$. Let $d$ be the least common multiple of $d_{j}'s$, i.e., $d = \operatorname{lcm}(d_{ 1}, \ldots, d_{ q})$. Then for any $\varepsilon > 0,$

where

with $l = (k+1)q!.$

2.   Notation and auxiliary results

2.1.   Some facts on holomorphic curves

To prove our result, we need the following second main theorem for holomorphic curves on the annulus (see ^[10,14]).

Lemma 1. (A general form of the second main theorem) Let $f: \mathbb{A} \rightarrow \mathbf{P}^{n}(\mathbb{C})$ be a linearly nondegenerate holomorphic curve (i.e. its image is not contained in any proper subspace of $\mathbf{P}^{n}(\mathbb{C})$). Let $H_{1}, \ldots, H_{q}$ (or linear forms $\mathbf{a}_{1}, \ldots, \mathbf{a}_{q}$) be arbitrary hyperplanes in $\mathbf{P}^{n}(\mathbb{C})$, then

Here, the maximum is taken over all subsets $K$ of $\{1, \ldots, q\}$ such that the linear forms $\mathbf{a}_{j}, j \in K$, are linearly independent.

We also need the following lemmas.

Lemma 2. (^[11,15]) Let $V$ be a projective subvariety of $\mathbf{P}^{n}(\mathbb{C})$ of dimension $k$. Let $D_{0}, \ldots, D_{p}$ be $p+1$ hypersurfaces in $\mathbf{P}^{n}(\mathbb{C})$ of the same degree $d \geq 1$, such that $\bigcap_{i = 0}^{p} D_{i}\cap V = \emptyset$ and

where $t_{0}, t_{1}, \ldots, t_{k}$ integers with $0 = t_{0} < t_{1} < \cdots < t_{k} = p$, then there exist $k+1$ hypersurfaces $P_{0}, \ldots, P_{k}$ in $\mathbf{P}^{n}(\mathbb{C})$ of the forms

such that $\left(\bigcap_{l = 0}^{k} P_{l}\right) \cap V = \emptyset$.

Lemma 3. (^[16]) Let $\left\{Q_{i}\right\}_{i \in R}$ be a set of hypersurfaces in $\mathbf{P}^{n}(\mathbb{C})$ of the common degree $d$, let $V$ be a projective subvariety of $\mathbf{P}^{n}(\mathbb{C})$ and let $f$ be a meromorphic mapping of ${\mathbb{C}}^{m}$ into $V$. Assume that $\bigcap_{i \in R} Q_{i} \cap V = \emptyset$, then there exist positive constants $\alpha$ and $\beta$ such that

Lemma 4. (^[11]) Let $t_{0}, t_{1}, \ldots, t_{n}$ be $n+1$ integers such that $1 = t_{0} < t_{1} < \cdots < t_{n}$, and let $\Delta = \max _{1 \leq s \leq n} \frac{t_{s}-t_{0}}{s}$, then for every $n$ real numbers $a_{0}, a_{1}, \ldots, a_{n-1}$ with $a_{0} \geq a_{1} \geq \cdots \geq a_{n-1} \geq 1$, we have

2.2.   Chow weights and Hilbert weights

We recall the notion of Chow weights and Hilbert weights from ^[5] (see also ^[17]). Let $X \subset \mathbf{P}^{n}(\mathbb{C})$ be a projective variety of dimension $k$ and degree $\delta$. The Chow form of $X$ is the unique polynomial, up to a constant scalar,

in $n+1$ blocks of variables $\mathbf{u}_{i} = \left(u_{i 0}, \ldots, u_{i n}\right), i = 0, \ldots, k$ with the following properties:

(ⅰ) $F_{X}$ is irreducible in $k\left[u_{00}, \ldots, u_{k n}\right]$;

(ⅱ) $F_{X}$ is homogeneous of degree $\delta$ in each block $\mathbf{u}_{i}, i = 0, \ldots, k$;

(ⅲ) $F_{X}\left(\mathbf{u}_{0}, \ldots, \mathbf{u}_{k}\right) = 0$, if and only if, $X \cap H_{\mathbf{u}_{0}} \cap \cdots \cap H_{\mathbf{u}_{k}} \neq \varnothing$, where $H_{\mathbf{u}_{i}}, i =$ $0, \ldots, k$, are the hyperplanes given by

Let $\mathbf{c} = \left(c_{0}, \ldots, c_{n}\right)$ be a tuple of real numbers and $t$ be an auxiliary variable. We consider the decomposition

with $G_{0}, \ldots, G_{r} \in \mathbb{C}\left[u_{00}, \ldots, u_{0 n}; \ldots; u_{k 0}, \ldots, u_{k n}\right]$ and $e_{0} > e_{1} > \cdots > e_{r}$. The Chow weight of $X$ with respect to $\mathbf{c}$ is defined by

For each subset $J = \left\{j_{0}, \ldots, j_{k}\right\}$ of $\{0, \ldots, n\}$ with $j_{0} < j_{1} < \cdots < j_{k}$, we define the bracket

where $\mathbf{u}_{i} = \left(u_{i 0}, \ldots, u_{i n}\right)(1 \leq i \leq k)$ denote the blocks of $n+1$ variables. Let $J_{1}, \ldots, J_{\beta}$ with $\beta = \left(\begin{array}{c}n+1 \\ k+1\end{array}\right)$ be all subsets of $\{0, \ldots, n\}$ of cardinality $k+1$.

Therefore, $F_{X}$ can be written as a homogeneous polynomial of degree $\delta$ in $\left[J_{1}\right], \ldots, \left[J_{\beta}\right]$. We may see that for $\mathbf{c} = \left(c_{0}, \ldots, c_{n}\right) \in \mathbb{R}^{n+1}$ and for any $J$ among $J_{1}, \ldots, J_{\beta}$,

For $\mathbf{a} = \left(a_{0}, \ldots, a_{n}\right) \in \mathbb{Z}^{n+1}$ we write $\mathbf{x}^{\mathbf{a}}$ for the monomial $x_{0}^{a_{0}} \cdots x_{n}^{a_{n}}$. Denote by $\mathbb{C}\left[x_{0}, \ldots, x_{n}\right]_{u}$ the vector space of homogeneous polynomials in $\mathbb{C}\left[x_{0}, \ldots, x_{n}\right]$ of degree $u$ (including 0). For an ideal $I$ in $\mathbb{C}\left[x_{0}, \ldots, x_{n}\right]$, we put

Let $I(X)$ be the prime ideal in $\mathbb{C}\left[x_{0}, \ldots, x_{n}\right]$ defining $X$. The Hilbert function $H_{X}$ of $X$ is defined by, for $u = 1, 2, \ldots$,

By the usual theory of Hilbert polynomials,

The $u$-th Hilbert weight $S_{X}(u, \mathbf{c})$ of $X$ with respect to the tuple $\mathbf{c} = \left(c_{0}, \ldots, c_{n}\right) \in$ $\mathbb{R}^{n+1}$ is defined by

where the maximum is taken over all sets of monomials $\mathbf{x}^{\mathbf{a}_{1}}, \ldots, \mathbf{x}^{\mathbf{a}_{H_{X}(u)}}$ whose residue classes modulo $I$ form a basis of $\mathbb{C}\left[x_{0}, \ldots, x_{n}\right]_{u} / I_{u}$. The following theorems are due to J. Evertse and R. Ferretti ^[18].

Lemma 5. Let $X \subset \mathbf{P}^{n}(\mathbb{C})$ be an algebraic variety of dimension $k$ and degree $\delta$. Let $u > \delta$ be an integer and let $\mathbf{c} = \left(c_{0}, \ldots, c_{n}\right) \in \mathbb{R}_{\geq 0}^{n+1}$, then

Lemma 6. Let $Y \subset \mathbf{P}^{n}(\mathbb{C})$ be an algebraic variety of dimension $k$ and degree $\delta.$ Let $\mathbf{c} = \left(c_{1}, \ldots, c_{q}\right)$ be a tuple of positive reals. Let $\left\{i_{0}, \ldots, i_{n}\right\}$ be a subset of $\{1, \ldots, q\}$ such that

then

3.   Proof of Theorem 1

Proof. Assume that $V$ is a projective subvariety of $\mathbf{P}^{n}(\mathbb{C})$ of dimension $k$. If there exists $i_{0}\in Q = \{1, 2, \ldots, q\}$ such that $\cap_{j \in Q\setminus{\{i_{0}\}}} D_{j}\bigcap V\neq\emptyset$, then it follows from the definition that

Hence, $q < \Delta_{V}(k+1)$, which implies the conclusion of Theorem 1 is trivial. Therefore, we only need to consider the case that for each $i\in Q$, the set $\cap_{j \in Q\setminus{\{i\}}} D_{j}\bigcap V = \emptyset.$

Replacing $D_{ j}$ by $D_{ j}^{d/d_{j}}$ if necessary, without loss of generality, we may assume that all hypersurfaces $D_{1}, \ldots, D_{q}$ are of the same degree $d$. We denote by $\{\sigma_{i}\}_{i\in I}$ the set of all permutations of the set $\{1, \ldots, q\}$, where $I = \{1, 2, \cdots, n_{0}\}$ and $n_{0} = q!$. For each $i \in I$, since $\bigcap_{j = 1}^{q-1} D_{\sigma_{i}(j)} \cap V = \emptyset$, there exist $k+1$ integers $t_{i, 0}, t_{i, 1}, \ldots, t_{i, k}$ with $1 = t_{i, 0} < \cdots < t_{i, k} = p_{i}$, where $p_{i} \leq q-1$ such that $\bigcap_{j = 1}^{p_{i}} D_{\sigma_{i}(j)}\cap V = \emptyset$ and

For each $i \in I$, we denote by $P_{i, 0}, \ldots, P_{i, k}$ the hypersurfaces obtained in Lemma 2 with respect to the hypersurfaces $D_{\sigma_{i}(1)}, \ldots, D_{\sigma_{i}(p_{i})}$.

We consider the mapping $\Phi$ from $V$ into $\mathbf{P}^{l}(\mathbb{C}) \; \left(l = n_{0}(k+1)-1\right)$, which maps a point $x \in V$ into the point $\Phi(x) \in \mathbf{P}^{l}(\mathbb{C})$ given by

Let $Y: = \Phi(V)$. Since $V \cap \left(\bigcap_{j = 0}^{k} P_{1, j}\right) = \emptyset, \Phi$ is a finite morphism on $V$ and $Y$ is a complex projective subvariety of $\mathbb{P}^{l}(\mathbf{C})$ with $\operatorname{dim} Y = k$ and

For every

and

we denote $\mathbf{y}^{\mathbf{a}} = y_{1, 0}^{a_{1, 0}} \cdots y_{1, k}^{a_{1, k}} \cdots y_{n_{0}, 0}^{a_{n_{0}, 0}} \cdots y_{n_{0}, k}^{a_{n_{0}, k}}$. Let $u$ be a positive integer. We set

and define the space

which is a vector space of dimension $H_{Y}(u)$. We fix a basis $\left\{v_{1}, \ldots, v_{H_{Y}(u)}\right\}$ of $Y_{u}$ and consider the meromorphic mapping $F$ with a reduced representation

Since $f$ is algebraically nondegenerate, the holomorphic curve $F:\mathbb{A} \to \mathbf{P}^{n_{u}}(\mathbb{C})$ is linearly nondegenerate (i.e., its image is not contained in any hyperplanes in $\mathbf{P}^{n_{u}}(\mathbb{C})$).

By Lemma 3, there exists a constant $A > 0$, which is chosen common for all $i \in I$, such that

According to the definition of $P_{i, j},$ we may choose a positive constant $B \geq 1$, commonly for all $i \in I$, such that

for all $\mathbf{x} = \left(x_{0}, \ldots, x_{n}\right) \in \mathbb{C}^{n+1}$ and for all $0 \leq j \leq k$. It is easily seen that, there exists a positive constant $C$, such that

for all $\mathbf{x} = \left(x_{0}, \ldots, x_{n}\right) \in \mathbb{C}^{n+1}, 1\leq i \leq n_{0}$, and $0 \leq j \leq k$.

Fix an element $i \in I$. Denote by $S(i)$ the set of all points

such that

Therefore, for each $z \in S(i)$, by Lemma 4 we have

which implies that

Now, we fix an index ${i} \in I$ and a point $z \in S(i)$ and define

where $c_{i, j, z}: = \log\frac{\| \tilde{f}(z)\|^{d} \|P_{i, j}\|}{\left|P_{i, j}(\tilde{f}(z))\right|}$ for $i = 1, \ldots, n_{0}$ and $j = 0, \ldots, k$. By the definition of the Hilbert weight, there are $\mathbf{a}_{1}, \ldots, \mathbf{a}_{H_{Y}(u)} \in \mathbb{Z}_{\geq 0}^{l+1}$ with

such that the residue classes modulo $\left(I_{Y}\right)_{u}$ of $\mathbf{y}^{\mathbf{a}_{1}, z}, \ldots, \mathbf{y}^{\mathbf{a}_{H_{Y}(u)}, z}$ forms a basis of $\mathbf{C}\left[y_{0}, \ldots, y_{l}\right]_{u} /\left(I_{Y}\right)_{u}$ and

Since $\mathbf{y}^{\mathbf{a}_{i}, z} \in Y_{u}$ (modulo $\left.\left(I_{Y}\right)_{u}\right)$, we may write

where $L_{i}\left(1 \leq i \leq H_{Y}(u)\right)$ are independent linear forms. We see at once that

This implies that

Here, we note that $L_{i, z}$ depends on $i$ and $z$, but the number of these linear forms is finite. We denote by $\mathcal{L}$ the set of all $L_{i, z}$ occurring in the above equalities, then we have

where the maximum is taken over all subsets $\mathcal{J} \subset \mathcal{L}$ with $\sharp \mathcal{J} = H_{Y}(u)$ and where $\{L; L \in \mathcal{J}\}$ is linearly independent. From Lemma 5, we have

We choose an index $i_{0}$ such that $z \in S\left(i_{0}\right)$. Since $P_{i_{0}, 1}, \ldots, P_{i_{0}, k+1}$ are in general with respect to $V$, by Lemma 6, we have

By combining (3.2), (3.3), and (3.4), we get

From (3.1) and (3.5), we have

where the term $O(1)$ does not depend on $z$. Integrating both sides of the above inequality, we then obtain

Applying Lemma 1 with $\epsilon^{\prime} > 0$ (which will be chosen later) to the holomorphic curve $F$ and linear forms $L_{i}\left(1 \leq i \leq H_{Y}(u)\right)$, we obtain that

Combining this inequality with (3.6), we have

We now estimate the quantity $N_{W(\tilde{F})}(r).$ In this case, we define

where $c_{i, j}: = \max\{\nu_{P_{i, j}(f)}(z)-n_{u}, 0\}$ for $i = 1, \ldots, n_{0}$ and $j = 0, \ldots, k$. By the definition of the Hilbert weight, there are $\mathbf{a}_{1}, \ldots, \mathbf{a}_{H_{Y}(u)} \in \mathbb{Z}_{\geq 0}^{l+1}$ with

such that the residue classes modulo $\left(I_{Y}\right)_{u}$ of $\mathbf{y}^{\mathbf{a}_{1}}, \ldots, \mathbf{y}^{\mathbf{a}_{H_{Y}}(u)}$ forms a basis of $\mathbb{C}\left[y_{0}, \ldots, y_{l}\right]_{u} /\left(I_{Y}\right)_{u}$ and

Again, there exist independent linear forms $\widehat{L_{i}}\left(1 \leq i \leq H_{Y}(u)\right)$ such that

We also easily see that

and hence

For the fixed point $z \in \triangle(R)$, without lose of generality, we may assume that

and $\sigma_{1} = (1, 2, \ldots, q)$.Since $P_{1, 0}, \ldots, P_{1, k}$ are in general position with respect to $V$, by Lemma 6 we have

This, together with Lemma 2, gives that

Note the definition of $P_{1, j} (0\leq j \leq k)$. We then have

Again, we set $t_{1, -1} = 0$. Thus, we derive from (3.9) that

Therefore, we derive from (3.8) and (3.10) that

Integrating both sides of this inequality, we obtain

Combining inequalities (3.7) and (3.11), we get

For each $\varepsilon > 0$, we now choose $u$ as the biggest integer such that

From (3.13) we have

Note that $\operatorname{deg} Y = \delta \leq d^{k} \operatorname{deg}(V)$,

Thus, it follows from (3.15) that

The proof of the theorem is finally completed. □

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was supported by Natural Science Research Project for Colleges and Universities of Anhui Province(Nos. 2022AH050329, 2022AH050290).

Conflict of interest

The authors declare there is no conflicts of interest.