Advantages that Starrk has against Ulquiorra: Superior reiatsu, faster reactions/reflexes, spammable attacks that can when launched at full-throttle can blast through Ulquiorra's Hierro and damage his internal organs beyond regeneration, the ability to move his attacks as he sees fit in response to attempts to evade from it.
Advantages that Ulquiorra has against Starrk: Has a superior Sonido that allows him to move at speeds faster than any Espada, is adept at manuevering in the air, can vaporize Starrk with Lanza.
Ulquiorra's moveset: Physical attacks, Lanza as a projectile, Lanza as a melee weapon, Cero Oscuras, Gran Ray Cero.
Starrk's moveset: Cero Metraletta, lupine warheads, Cero Oscuras, Gran Ray Cero, Colmillo.
Lupine warheads are able to hound Ulquiorra down and scatter his body parts or deal sufficient internal damage upon accumulated explosions.
Cero Metraletta is able to nuke Ulquiorra, and Starrk can move his arm/gun around in case Ulquiorra tries to dodge it. CO can also be used to overpower CO, and counter Lanza by making it explode prematurely (although this would result in both of their deaths).
Lanza has minimal accuracy and someone with reflexes and skill in evasion like Starrk would never get hit by it. Ulquiorra also tends to throw it above shoulder-height, meaning if it hits someone, it would hit them right in the head or neck area. That's a small surface area to aim at, thus making it even easier to dodge Lanza.
Segunda Etapa's power boost is speculative, and based on what's stated and shown, the only difference between R2 and R1 is that R2 increases Ulquiorra's AoE and destructive output via Lanza as well as changing the nature of his reiatsu to instill a sense of despair and hopelessness in others. This change in nature makes Ulquiorra's reiatsu very dense, to the point that someone sensing it will be under the illusion that it is like an ocean above the sky, when they're really falling spell to the despair nature because they don't know exactly what it is. This is similar to Ulquiorra and Hallibel misjudging Vizard ichigo to be an Espada based on the feeling that the nature of his reiatsu gives off.
Therefore, the best hypothesis is that Ulquiorra does not gain any more reiatsu than he did in the first place, meaning even Hallibel technically has more reiatsu than him still. If he were the most powerful Espada, he would have informed Aizen, who he swore absolute loyalty to, about a flaw in the rankings.
Starrk, even before he became an Espada, possessed a reiatsu so strong that it unconsciously killed dozens of Hollows near him. This feat impressed Aizen enough that he made Starrk the Primera. Furthermore, even after splitting 35+ pieces off of his soul for his wolves, he did not show any signs of being taxed from excessive reiatsu/reiryoku loss. Furthermore, he was even able to fight on a comparable level against a serious Shunsui in swordsmanship despite losing so much reiatsu, and even nearly blitzed him with his Sonido. All without using instant Ceros, CO, or GRC. This all attests to the volume of Starrk's overall reiatsu levels.
So Starrk has superior reiatsu and reactions than Ulquiorra, and his abilities and skills are a good-match up against his, therefore he wins more often than not.
Fri Feb 20, 2015 1:21 pm