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The integer $c_1^2(S) - c_2(S)$ is the second Segre class of the surface $S$. For surfaces of general type a Riemann-Roch computation shows that its positivity implies that $$\lim \sup \frac{\log h^0\big(S,\mathrm{Sym}^i(\Omega^1_S) \otimes \mathcal L\big) }{\log i} > 0 =3$$ for any line-bundle $\mathcal L$.

This fact has been explored by Bogomolov back in the seventies, to prove the finiteness of rational and elliptic curves on surfaces of general type with positive second Segre class, see for instance this Bourbaki seminar. The point of Bogomolov's argument is that a symmetric $1$-form defines a multi-foliation (also called web) on $S$. If $i: \mathbb P^1 \to S$ is a non-trivial morphism and $\omega \in H^0(S,\mathrm{Sym}^i \Omega^1_S)$ then $$i^* \omega \in H^0(\mathbb P^1, \mathrm{Sym}^i \Omega^1_{\mathbb P^1}) = H^0(\mathbb P^1,\mathcal O_{\mathbb P^1}(-2i)) .$$ We deduce that $i^* \omega$ vanishes identically, i.e., the image of $i$ is a leaf of the multi-foliation defined by $\omega$. If there are infinitely many of them, a theorem by Jouanolou implies that we have a $1$-parameter family of rational curves on $S$, thus $S$ is uniruled and cannot be of general type. If we start with a section of $\mathrm{Sym}^i \Omega^1_S \otimes \mathcal L$ with $\mathcal L^*$ ample, the very same argument shows the finiteness of elliptic curves on $S$. A more involved argument, but following the same lines, shows the boundeness of curves of bounded genus.

More recently, McQuillan proved that surfaces of general type with positive sencond Segre class do not admit Zariski dense entire curves in Diophantine approximations and foliations. This work lead to a birational classification of foliations on projective surfaces (by McQuillan, Brunella, and Mendes) very much in the spirit of Enriques-Kodaira classification, see this paper and references therein.

Similar results are not known for surfaces satisfying the inequality you impose: $c_2 -c_1^2\ge 0$. Already the starting point, the existence of symmetric differential forms, is rather non-trivial. Very recently Demailly proved the existence higher order differential equations with coefficients in duals of ample line-bundles.

Concerning surfaces with non-positive second Segre class let me mention that any smooth surface in $\mathbb P^3$ of degree at least $5$ is of general type and satisfies the inequality $c_1^2(S) - c_2(S) \le 0$. As far as I know, even the finiteness of rational curves on a generic quintic surface is unknown.

If, by any chance, one knows that they are minimal then the graph below borrowed from this wikipedia page might tell something. Notice that the line contained containing the ruled surfaces is $2c_2 = c_1^2$.

Otherwise the situation is not very even less encouraging, when you . After a blow-up $c_2$ increases by one while $c_1^2$ decreases by one. After sufficiently many blow-ups you we always end up with a surface with negative second Segre class.

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The integer $c_1^2(S) - c_2(S)$ is the second Segre class of the surface $S$. For surfaces of general type a Riemann-Roch computation shows that its positivity implies that $$\lim \sup \frac{\log h^0\big(S,\mathrm{Sym}^i(\Omega^1_S) \otimes \mathcal L\big) }{\log i} > 0$$ for any line-bundle $\mathcal L$.

This fact has been explored by Bogomolov back in the seventies, to prove the finiteness of rational and elliptic curves on surfaces of general type with positive second Segre class, see for instance this Bourbaki seminar. The point of Bogomolov's argument is that a symmetric $1$-form defines a multi-foliation (also called web) on $S$. If $i: \mathbb P^1 \to S$ is a non-trivial morphism and $\omega \in H^0(S,\mathrm{Sym}^i \Omega^1_S)$ then $$i^* \omega \in H^0(\mathbb P^1, \mathrm{Sym}^i \Omega^1_{\mathbb P^1}) = H^0(\mathbb P^1,\mathcal O_{\mathbb P^1}(-2i)) .$$ We deduce that $i^* \omega$ vanishes identically, i.e., the image of $i$ is a leaf of the multi-foliation defined by $\omega$. If there are infinitely many of them, a theorem by Jouanolou implies that we have a $1$-parameter family of rational curves on $S$, thus $S$ is uniruled and cannot be of general type. If we start with a section of $\mathrm{Sym}^i \Omega^1_S \otimes \mathcal L$ with $\mathcal L^*$ ample, the very same argument shows the finiteness of elliptic curves on $S$. A more involved argument, but following the same lines, shows the boundeness of curves of bounded genus.

More recently, McQuillan proved that surfaces of general type with positive sencond Segre class do not admit Zariski dense entire curves in Diophantine approximations and foliations. This work lead to a birational classification of foliations on projective surfaces (by McQuillan, Brunella, and Mendes) very much in the spirit of Enriques-Kodaira classification, see this paper and references therein.

Similar results are not known for surfaces satisfying the inequality you impose: $c_2 -c_1^2\ge 0$. Already the starting point, the existence of symmetric differential forms, is rather non-trivial. Very recently Demailly proved the existence higher order differential equations with coefficients in duals of ample line-bundles.

Concerning surfaces with non-positive second Segre class let me mention that any smooth surface in $\mathbb P^3$ of degree at least $5$ is of general type and satisfies the inequality $c_1^2(S) - c_2(S) \le 0$. As far as I know, even the finiteness of rational curves on a generic quintic surface is unknown.

If, by any chance, one knows that they are minimal then the graph below borrowed from this wikipedia page might tell something. Notice that the line contained the ruled surfaces is $2c_2 = c_1^2$.

Otherwise the situation is not very encouraging, when you blow-up $c_2$ increases by one while $c_1^2$ decreases by one. After sufficiently many blow-ups you always end up with a surface with negative second Segre class.

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The integer $c_1^2(S) - c_2(S)$ is the second Segre class of the surface $S$. For surfaces of general type a Riemann-Roch computation shows that its positivity implies that $$\lim \sup \frac{\log h^0\big(S,\mathrm{Sym}^i(\Omega^1_S) \otimes \mathcal L\big) }{\log i} > 0$$ for any line-bundle $\mathcal L$.

This fact has been explored by Bogomolov back in the seventies, to prove the finiteness of rational and elliptic curves on surfaces of general type with positive second Segre class, see for instance this Bourbaki seminar. The point of Bogomolov's argument is that the a symmetric $1$-form defines a multi-foliation (also called web) on $S$. If $i: \mathbb P^1 \to S$ is a non-trivial morphism and $\omega \in H^0(S,\mathrm{Sym}^i \Omega^1_S)$ then $$i^* \omega \in H^0(\mathbb P^1, \mathrm{Sym}^i \Omega^1_{\mathbb P^1}) = H^0(\mathbb P^1,\mathcal O_{\mathbb P^1}(-2i)) .$$ We deduce that $i^* \omega$ vanishes identically, i.e., the image of $i$ is a leaf of the multi-foliation defined by $\omega$. If there are infinitely many of them, a theorem by Jouanolou implies that we have a $1$-parameter family of rational curves on $S$, thus $S$ is uniruled and cannot be of general type. If we start with a section of $\mathrm{Sym}^i \Omega^1_S \otimes \mathcal L$ with $\mathcal L^*$ ample, the very same argument shows the finiteness of elliptic curves on $S$. A more involved argument, but following the same lines, shows the boundeness of curves of bounded genus.

More recently, McQuillan proved that surfaces of general type with positive sencond Segre class do not admit Zariski dense entire curves in Diophantine approximations and foliations. This work lead to a birational classification of foliations on projective surfaces (by McQuillan, Brunella, and Mendes) very much in the spirit of Enriques-Kodaira classification, see for this paper and references therein.

Similar results are not known for surfaces satisfying the inequality you impose: $c_2 -c_1^2\ge 0$. Already the starting point, the existence of symmetric differential forms, is rather non-trivial. Very recently Demailly proved the existence higher order differential equations with coefficients in duals of ample line-bundles.

As in the body of the question you ask about surfaces with non-positive second Segre class let me mentions mention that any smooth surface in $\mathbb P^3$ of degree at least $5$ is of general type and satisfies the inequality $c_1^2(S) - c_2(S) \le 0$. As far as I know, even the finiteness of rational curves on a generic quintic surface is unknown.

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